**Application of Harmony Search Algorithm in Power Engineering**

H. R. Baghaee, M. Mirsalim and G. B. Gharehpetian

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/55509

## **1. Introduction**

## **1.1. On the use of harmony search algorithm in optimal placement of FACTS devices to improve power systems**

With the increasing electric power demand, power systems can face to stressed conditions, the operation of power system becomes more complex, and power system will become less secure. Moreover, because of restructuring, the problem of power system security has become a matter of concern in deregulated power industry. Better utilization of available power system capacities by Flexible AC Transmission Systems (FACTS) devices has become a major concern in power systems too.

FACTS devices can control power transmission parameters such as series impedance, voltage, and phase angle by their fast control characteristics and continuous compensating capability. They can reduce flow of heavily loaded lines, resulting in low system losses, improved both transient and small signal stability of network, reduced cost of production, and fulfillment of contractual requirement by controlling the power flow in the network. They can enable lines to flow the power near its nominal rating and maintain its voltage at desired level and thus, enhance power system security in contingencies [1-6]. For a meshed network, an optimal allocation of FACTS devices allows to control its power flows and thus, to improve the system loadability and security [1].

The effect of FACTS devices on power system security, reliability and loadability has been studied according to proper control objectives [4-14]. Researchers have tried to find suitable location for FACTS devices to improve power system security and loadability [13-16]. The optimal allocation of these devices in deregulated power systems has been presented in

© 2013 Baghaee et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Baghaee et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

[17-18]. Heuristic approaches and intelligent algorithms to find suitable location of FACTS devices and some other applications have been used in [15-21].

In this chapter, a novel heuristic method is presented based on Harmony Search Algorithm (HSA) to find optimal location of multi-type FACTS devices to enhance power system security and reduce power system losses considering investment cost of these devices. The proposed method is tested on IEEE 30-bus system and then, the results are presented.

## **2. Model of FACTS devices**

## **2.1. FACTS devices**

In this chapter, we select three different FACTS devices to place in the suitable locations to improve security margins of power systems. They are TCSC (Thyristor Controlled Series Capacitor), SVC (Static VAR Compensator), and UPFC (Unified Power Flow Controller) that are shown in Fig. 1.

Power flow through the transmission line i-j namely *Pij*, depends on the line reactance *Xij*, the bus voltage magnitudes *Vi* , and *Vj* , and phase angle between sending and receiving buses *δ<sup>i</sup>* and *δ<sup>j</sup>* , expressed by Eq. 1.

$$P\_{i\dot{j}} = \frac{V\_i V\_j}{X\_{i\dot{j}}} \sin(\mathcal{S}\_i - \mathcal{S}\_j) \tag{1}$$

*XX X ij line TCSC* = + (2)

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203

. *X rX TCSC TCSC line* = (3)

*Q Q i SVC* D = (4)

where, *Xline* is the reactance of the transmission line, and *rTCSC* is the compensation factor of TCSC. The rating of TCSC depends on transmission line. To prevent overcompensation, we

SVC can be used for both inductive and capacitive compensation. In this chapter, SVC is

Two types of UPFC models have been studied in the literature; one is the coupled model [28], and the other the decoupled type[29-31]. In the first, UPFC is modeled with series combination

choose TCSC reactance between −0.7*Xline* to 0.2*Xline* [26-27].

modeled as an ideal reactive power injection at bus *i*:

**Figure 1.** Models of FACTS Devices (a) TCSC, (b) SVC and (c) UPFC

*2.2.2. SVC*

*2.2.3. UPFC*

TCSC can change line reactance, and SVC can control the bus voltage. UPFC is the most versatile member of FACTS devices family and controls all power transmission parameters (i.e., line impedance, bus voltage, and phase angles). FACTS devices can control and optimize power flow by changing power system parameters. Therefore, optimal device and allocation of FACTS devices can result in suitable utilization of power systems.

### **2.2. Mathematical model of FACTS devices**

In this chapter, steady-state model of FACTS devices are developed for power flow studies. TCSC is simply modeled to modify just the reactance of transmission lines. SVC and UPFC are modeled using the power injection models. Therefore, SVC is modeled as shunt element of transmission line, and UPFC as decoupled model. A power flow program has been developed in MATLAB by incorporating the mathematical models of FACTS devices.

## *2.2.1. TCSC*

TCSC compensates the reactance of the transmission line. This changes the line flow due to change in series reactance. In this chapter, TCSC is modeled by changing transmission line reactance as follows:

Application of Harmony Search Algorithm in Power Engineering http://dx.doi.org/10.5772/55509 203

**Figure 1.** Models of FACTS Devices (a) TCSC, (b) SVC and (c) UPFC

$$X\_{ij} = X\_{line} + X\_{TSCO} \tag{2}$$

$$X\_{\text{TCSC}} = r\_{\text{TCSC}} X\_{\text{line}} \tag{3}$$

where, *Xline* is the reactance of the transmission line, and *rTCSC* is the compensation factor of TCSC. The rating of TCSC depends on transmission line. To prevent overcompensation, we choose TCSC reactance between −0.7*Xline* to 0.2*Xline* [26-27].

*2.2.2. SVC*

[17-18]. Heuristic approaches and intelligent algorithms to find suitable location of FACTS

In this chapter, a novel heuristic method is presented based on Harmony Search Algorithm (HSA) to find optimal location of multi-type FACTS devices to enhance power system security and reduce power system losses considering investment cost of these devices. The proposed

In this chapter, we select three different FACTS devices to place in the suitable locations to improve security margins of power systems. They are TCSC (Thyristor Controlled Series Capacitor), SVC (Static VAR Compensator), and UPFC (Unified Power Flow Controller) that

Power flow through the transmission line i-j namely *Pij*, depends on the line reactance *Xij*, the

sin( ) *i j ij i j ij*

d d

TCSC can change line reactance, and SVC can control the bus voltage. UPFC is the most versatile member of FACTS devices family and controls all power transmission parameters (i.e., line impedance, bus voltage, and phase angles). FACTS devices can control and optimize power flow by changing power system parameters. Therefore, optimal device and allocation

In this chapter, steady-state model of FACTS devices are developed for power flow studies. TCSC is simply modeled to modify just the reactance of transmission lines. SVC and UPFC are modeled using the power injection models. Therefore, SVC is modeled as shunt element of transmission line, and UPFC as decoupled model. A power flow program has been developed

TCSC compensates the reactance of the transmission line. This changes the line flow due to change in series reactance. In this chapter, TCSC is modeled by changing transmission line

*V V*

, and phase angle between sending and receiving buses *δ<sup>i</sup>*


devices and some other applications have been used in [15-21].

, and *Vj*

*P <sup>X</sup>* <sup>=</sup>

of FACTS devices can result in suitable utilization of power systems.

in MATLAB by incorporating the mathematical models of FACTS devices.

**2.2. Mathematical model of FACTS devices**

**2. Model of FACTS devices**

202 Search Algorithms for Engineering Optimization

**2.1. FACTS devices**

are shown in Fig. 1.

and *δ<sup>j</sup>*

*2.2.1. TCSC*

reactance as follows:

bus voltage magnitudes *Vi*

, expressed by Eq. 1.

method is tested on IEEE 30-bus system and then, the results are presented.

SVC can be used for both inductive and capacitive compensation. In this chapter, SVC is modeled as an ideal reactive power injection at bus *i*:

$$
\Delta Q\_i = Q\_{SVC} \tag{4}
$$

#### *2.2.3. UPFC*

Two types of UPFC models have been studied in the literature; one is the coupled model [28], and the other the decoupled type[29-31]. In the first, UPFC is modeled with series combination of a voltage source and impedance in the transmission line. In the decoupled model, UPFC is modeled with two separated buses. The first model is more complex than the second one because the modification of the Jacobian matrix is inevitable. In conventional power flow algorithms, we can easily implement the decoupled model. In this chapter, the decoupled model has been used to model the UPFC as in Fig. 2.

UPFC controls power flow of the transmission lines. To present UPFC in load flow studies, the variables *Pu*1, *Qu*1, *Pu*2, and *Qu*<sup>2</sup> are used. Assuming a lossless UPFC, real power flow from bus *i* to bus *j* can be expressed as follows:

$$P\_{ij} = P\_{u1} \tag{5}$$

2

*J w <sup>S</sup>* <sup>=</sup> å (8)

Application of Harmony Search Algorithm in Power Engineering

*th* line or transformer.*JP* is the security index for the even

*th* bus, respectively,

*th* line, respectively,*Vref* ,*<sup>i</sup>*

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205

, and *X* is the set of the possible range

, ( ) *<sup>j</sup>*

as nominal voltage magnitude, which is assumed to be 1*pu* for all load buses (i.e., PQ bus‐ es), and to be equal to specified value for generation buses (i.e., PV buses), and*S <sup>j</sup>*,max as

distribution of the total active flow, and *JV* is the security index for the closeness of bus volt‐ age to the reference voltage. If the number of overloaded lines decreases, the value of *JP* reduces too. Similarly, when the bus voltage have a value close to the desired level, *JV* be‐ comes a small value. Minimization of both *JP* and *JV* means the maximization of security

Harmony Search Algorithm (HSA) has recently been developed in an analogy with music improvisation process, where music players improvise the pitches of their instruments to obtain better harmony [34]. The steps in the procedure of harmony search are as follows [35]:

where, *f(x)* is the objective function, *g(x)* the inequality constraint function, and *h(x)* the equality

of values for each decision variable; that is *Xi*,min ≤ *Xi* ≤ *Xi*,max where, *Xi*,min and *Xi*,max are the lower and upper bounds of each decision variable. The HS algorithm parameters are also

*S*

*j j amx*

*P j*

as voltage amplitude and associated weighting factor for the *i*

as apparent power and associated weighting factor for the *j*

Here we have:

nominal apparent power of the *j*

**4. The proposed algorithm**

**4.1. Harmony search algorithm**

**Step 2**: Initialize the harmony memory

**Step 4**: Update the harmony memory **Step 5**: Check the stopping criterion

**Step 3**: Improvise a new harmony

**Step 1**: Initialize the problem and algorithm parameters

The next following five subsections describe these steps.

In step 1, the optimization problem is specified as follows:

constraint function. *x* is the set of each decision variable *xi*

**a.** Initialize the problem and algorithm parameters

min { *f* (*x*)| *x* ∈ *X* }subject to *g*(*x*)≥0 and *h* (*x*)=0

*Vi* , *wi*

*Sj* , *wj*

margins.

Although UPFC can control the power flow but, it cannot generate the real power. Therefore, we have:

$$P\_{\mu1} + P\_{\mu2} = 0 \tag{6}$$

Reactive power output of UPFC, *Qu*1 , and *Qu*2, can be set to an arbitrary value depending on the rating of UPFC to maintain bus voltage.

**Figure 2.** Decoupled model for UPFC

### **3. Security index**

The security index for contingency analysis of power systems can be expressed as in the following [32-33]:

$$J\_V = \sum\_i w\_i \left| V\_i - V\_{ref,i} \right|^2 \tag{7}$$

Application of Harmony Search Algorithm in Power Engineering http://dx.doi.org/10.5772/55509 205

$$J\_P = \sum\_j w\_j (\frac{\mathbf{S}\_j}{\mathbf{S}\_{j,\text{max}}})^2 \tag{8}$$

Here we have:

of a voltage source and impedance in the transmission line. In the decoupled model, UPFC is modeled with two separated buses. The first model is more complex than the second one because the modification of the Jacobian matrix is inevitable. In conventional power flow algorithms, we can easily implement the decoupled model. In this chapter, the decoupled

UPFC controls power flow of the transmission lines. To present UPFC in load flow studies, the variables *Pu*1, *Qu*1, *Pu*2, and *Qu*<sup>2</sup> are used. Assuming a lossless UPFC, real power flow from

Although UPFC can control the power flow but, it cannot generate the real power. Therefore,

Reactive power output of UPFC, *Qu*1 , and *Qu*2, can be set to an arbitrary value depending on

The security index for contingency analysis of power systems can be expressed as in the

*V i i ref i*, *i*

2

*J wV V* = - å (7)

*ij u*<sup>1</sup> *P P* = (5)

1 2 0 *u u P P* + = (6)

model has been used to model the UPFC as in Fig. 2.

bus *i* to bus *j* can be expressed as follows:

204 Search Algorithms for Engineering Optimization

the rating of UPFC to maintain bus voltage.

**Figure 2.** Decoupled model for UPFC

**3. Security index**

following [32-33]:

we have:

*Vi* , *wi* as voltage amplitude and associated weighting factor for the *i th* bus, respectively, *Sj* , *wj* as apparent power and associated weighting factor for the *j th* line, respectively,*Vref* ,*<sup>i</sup>* as nominal voltage magnitude, which is assumed to be 1*pu* for all load buses (i.e., PQ bus‐ es), and to be equal to specified value for generation buses (i.e., PV buses), and*S <sup>j</sup>*,max as nominal apparent power of the *j th* line or transformer.*JP* is the security index for the even distribution of the total active flow, and *JV* is the security index for the closeness of bus volt‐ age to the reference voltage. If the number of overloaded lines decreases, the value of *JP* reduces too. Similarly, when the bus voltage have a value close to the desired level, *JV* be‐ comes a small value. Minimization of both *JP* and *JV* means the maximization of security margins.

## **4. The proposed algorithm**

#### **4.1. Harmony search algorithm**

Harmony Search Algorithm (HSA) has recently been developed in an analogy with music improvisation process, where music players improvise the pitches of their instruments to obtain better harmony [34]. The steps in the procedure of harmony search are as follows [35]:

**Step 1**: Initialize the problem and algorithm parameters

**Step 2**: Initialize the harmony memory

**Step 3**: Improvise a new harmony

**Step 4**: Update the harmony memory

**Step 5**: Check the stopping criterion

The next following five subsections describe these steps.

**a.** Initialize the problem and algorithm parameters

In step 1, the optimization problem is specified as follows:

min { *f* (*x*)| *x* ∈ *X* }subject to *g*(*x*)≥0 and *h* (*x*)=0

where, *f(x)* is the objective function, *g(x)* the inequality constraint function, and *h(x)* the equality constraint function. *x* is the set of each decision variable *xi* , and *X* is the set of the possible range of values for each decision variable; that is *Xi*,min ≤ *Xi* ≤ *Xi*,max where, *Xi*,min and *Xi*,max are the lower and upper bounds of each decision variable. The HS algorithm parameters are also specified in this step. These are the harmony memory size (HMS), or the number of solution vectors in the harmony memory, harmony memory considering rate (HMCR), pitch adjusting rate (PAR), the number of decision variables (N), and the number of improvisations (NI), or stopping criterion. The harmony memory (HM) is a memory location where all the solution vectors (sets of decision variables) are stored. This HM is similar to the genetic pool in the GA [36]. Here, HMCR and PAR are parameters that are used to improve the solution vector and are defined in step 3.

#### **b.** Initialize the harmony memory

In step 2, the HM matrix is filled with as many randomly generated solution vectors as the HMS in the following:

$$HM = \begin{bmatrix} \mathbf{x}\_1^1 & \mathbf{x}\_2^1 & \dots & \mathbf{x}\_{N-1}^1 & \mathbf{x}\_N^1\\ \mathbf{x}\_1^2 & \mathbf{x}\_2^2 & \dots & \mathbf{x}\_{N-1}^2 & \mathbf{x}\_N^2\\ \vdots & \vdots & \vdots & \vdots & \vdots\\ \mathbf{x}\_1^{HMS-1} & \mathbf{x}\_2^{HMS-1} & \dots & \mathbf{x}\_{N-1}^{HMS-1} & \mathbf{x}\_N^{HMS-1}\\ \mathbf{x}\_1^{HMS} & \mathbf{x}\_2^{HMS} & \dots & \mathbf{x}\_{N-1}^{HMS} & \mathbf{x}\_N^{HMS} \end{bmatrix} \tag{9}$$

*x* ′ *i*

"Yes", *x* ′

*<sup>i</sup>* ± *rand*()\* *bw*

variable of the new harmony vector.

**d.** Update harmony memory

If the new harmony vector *x* ′

**e.** Check stopping criterion

**4.2. Cost of FACTS devices**

as follows:

For TCSC:

For SVC:

For UPFC:

*x* ′ *<sup>i</sup>* ← *x* ′

and 1.

(1 )

<sup>ï</sup> - <sup>î</sup> (11)

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*<sup>N</sup>* ) is better than the worst harmony in the HM,

*<sup>i</sup>* is

207

*Yes with probability PAR No with probability PAR*

The value of (1-PAR) sets the rate of doing nothing. If the pitch adjustment decision for *x* ′

where, *bw* is an arbitrary distance bandwidth and rand () is a random number between 0

In step 3, HM consideration, pitch adjustment or random selection in turn is applied to each

judged in terms of the objective function value, the new harmony is included in the HM, and

If the stopping criterion (maximum number of improvisations) is satisfied, the computation

Using database of [32], cost function for SVC, TCSC, and UPFC shown in Fig. 3 are modeled

<sup>2</sup> 0.0015 0.713 153.75 *C ss TCSC* = -+ (12)

<sup>2</sup> 0.0003 0.3015 127.38 *C ss SVC* = -+ (13)

<sup>2</sup> 0.0003 0.2691 188.22 *C ss UPFC* = -+ (14)

2, ..., *x* ′

ìï ¬ í

*<sup>i</sup>* =(*x* ′ 1, *x* ′

the existing worst harmony is excluded from the HM.

terminates. Otherwise, steps 3, and 4 are repeated.

*<sup>i</sup>* will be replaced as follows:

#### **c.** Improvise a new harmony

range of values.

A new harmony vector, *x* ′ *<sup>i</sup>* =(*x* ′ 1, *x* ′ 2, ..., *x* ′ *<sup>N</sup>* ) , is generated based on three rules; (1) memory consideration, (2) pitch adjustment, and (3) random selection. Generating a new harmony is called 'improvisation' [36]. In the memory consideration, the value of the first decision variable *x* ′ 1 for the new vector is chosen from any value in the specified HM range (*x* 1 <sup>1</sup> − *x*<sup>1</sup> *HMS* ). Values of the other decision variables (*x* ′ 2, *x* ′ 3, ..., *x* ′ *<sup>N</sup>* ), are chosen in the same manner. The HMCR, which varies between zero and one, is the rate of choosing one value from the historical values stored in the HM, while (1-HMCR) is the rate of randomly selecting one value from the possible

$$\mathbf{x}\_{i}^{\prime} \leftarrow \begin{cases} \mathbf{x}\_{i}^{\prime} \in \left\{ \mathbf{x}\_{i}^{1}, \mathbf{x}\_{i}^{2}, \dots, \mathbf{x}\_{i}^{HMS} \right\} \text{ with probability HMCR} \\ \mathbf{x}\_{i}^{\prime} \in \mathbf{X}\_{i} & \text{with probability (1 - HMCR)} \end{cases} \tag{10}$$

For example, an HMCR of 0.85 indicates that the HS algorithm will choose the decision variable value from historically stored values in the HM with 85% probability or from the entire possible range with (100–85) % probability. Every component obtained by the memory consideration is examined to determine whether it should be pitch-adjusted. This operation uses the PAR parameter, which is the rate of pitch adjustment as follows:

*x* ′ *i*

specified in this step. These are the harmony memory size (HMS), or the number of solution vectors in the harmony memory, harmony memory considering rate (HMCR), pitch adjusting rate (PAR), the number of decision variables (N), and the number of improvisations (NI), or stopping criterion. The harmony memory (HM) is a memory location where all the solution vectors (sets of decision variables) are stored. This HM is similar to the genetic pool in the GA [36]. Here, HMCR and PAR are parameters that are used to improve the solution vector and

In step 2, the HM matrix is filled with as many randomly generated solution vectors as the

11 1 1 12 1 22 2 2 12 1

... ...

é ù ê ú

*xx x x xx x x*

> ... ...

*HMS HMS HMS HMS*

ë û

consideration, (2) pitch adjustment, and (3) random selection. Generating a new harmony is called 'improvisation' [36]. In the memory consideration, the value of the first decision variable

which varies between zero and one, is the rate of choosing one value from the historical values stored in the HM, while (1-HMCR) is the rate of randomly selecting one value from the possible

*x x x x with probability HMCR*

*x X with probability HMCR*

For example, an HMCR of 0.85 indicates that the HS algorithm will choose the decision variable value from historically stored values in the HM with 85% probability or from the entire possible range with (100–85) % probability. Every component obtained by the memory consideration is examined to determine whether it should be pitch-adjusted. This operation uses the PAR

*HMS HMS HMS HMS*

12 1 12 1

2, ..., *x* ′

1 for the new vector is chosen from any value in the specified HM range (*x*

3, ..., *x* ′

*HMS*

ï ¢ Î - î

2, *x* ′

{ } 1 2 , ,...,

*i ii i*

*i i*

parameter, which is the rate of pitch adjustment as follows:

*xx xx xx xx*

11 11


*N N N N*

M MMM M (9)

*<sup>N</sup>* ) , is generated based on three rules; (1) memory

*<sup>N</sup>* ), are chosen in the same manner. The HMCR,

(1 )

1 <sup>1</sup> − *x*<sup>1</sup> *HMS* ). Values

(10)


*N N*

*N N*

are defined in step 3.

HMS in the following:

**b.** Initialize the harmony memory

206 Search Algorithms for Engineering Optimization

*HM*

**c.** Improvise a new harmony

of the other decision variables (*x* ′

*i*

ì <sup>ï</sup> ¢ <sup>Î</sup> ¢ <sup>¬</sup> <sup>í</sup>

*x*

A new harmony vector, *x* ′

*x* ′

range of values.

=

*<sup>i</sup>* =(*x* ′ 1, *x* ′

$$
\epsilon \leftarrow \begin{cases}
\text{Yes} & \text{with probabilityPAR} \\
\text{No} & \text{with probability} (1 - PAR)
\end{cases}
\tag{11}
$$

The value of (1-PAR) sets the rate of doing nothing. If the pitch adjustment decision for *x* ′ *<sup>i</sup>* is "Yes", *x* ′ *<sup>i</sup>* will be replaced as follows:

$$\text{ax}'\_{i} \leftarrow \text{ax}'\_{i} \pm rand(\text{)\*} b\_w$$

where, *bw* is an arbitrary distance bandwidth and rand () is a random number between 0 and 1.

In step 3, HM consideration, pitch adjustment or random selection in turn is applied to each variable of the new harmony vector.

**d.** Update harmony memory

If the new harmony vector *x* ′ *<sup>i</sup>* =(*x* ′ 1, *x* ′ 2, ..., *x* ′ *<sup>N</sup>* ) is better than the worst harmony in the HM, judged in terms of the objective function value, the new harmony is included in the HM, and the existing worst harmony is excluded from the HM.

**e.** Check stopping criterion

If the stopping criterion (maximum number of improvisations) is satisfied, the computation terminates. Otherwise, steps 3, and 4 are repeated.

#### **4.2. Cost of FACTS devices**

Using database of [32], cost function for SVC, TCSC, and UPFC shown in Fig. 3 are modeled as follows:

For TCSC:

$$C\_{T\text{CCS}} = 0.0015\text{s}^2 - 0.713\text{s} + 153.75\tag{12}$$

For SVC:

$$\mathcal{L}\_{\text{SVC}} = 0.0003 \text{s}^2 - 0.3015 \text{s} + 127.38 \tag{13}$$

For UPFC:

$$\mathcal{C}\_{\text{LIPFC}} = 0.0003 \text{s}^2 - 0.2691 \text{s} + 188.22 \tag{14}$$

Here, *s* is the operating range of the FACTS devices in MVAR, and *CTCSC*, *CSVC*, and *CUPFC* are in *\$US* / *kVAR*.

and SVC, described in equations (12) to (14). In addition, the forth term is total active power

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209

Simulation studies are carried out for different scenarios in the IEEE 30-bus power system.

The first case is the normal operation of network without using any FACTS device. In the second, third, and fourth cases, installation of only one device has been considered. Each device is placed in an optimal location obtained by HSA. Multi-type FACTS devices installation is considered in the 5th scenario. In this case, three different kinds of FACTS devices have been

The performance index evolutions of implemented methods are shown in Fig.4, and Fig.5. The average, and maximum performance indices are shown in Fig.4. Tables 1, and 2 show optimal locations of devices for different cases. These results illustrate that the installation of one device in the network could not lead to improved security of power system and reduction in power system losses simultaneously, and that multi-type FACTS devices should be placed in optimal

> **Size (MVA)**

TCSC - - 90.6 1-2 - -

SVC - - - - 39 1

UPFC 48.3 12-15 - - - -

Multi-type 75.9 12-15 73.1 2-5 66.7 1

**UPFC TCSC SVC**

**Location (Bus No-Bus No)**

**Size (MVA)**

**Location (Bus No.)**

**•** Case 1: power system normal operation (without installation of FACTS devices),

**•** Case 5: Multi-type (TCSC, SVC, and UPFC) FACTS devices are installed.

considered to be placed in optimal locations to enhance power system security.

locations to improve security margins and reduce losses in the network.

**Location (Bus No-Bus No)**

loss of the power system.

**5. Simulation results**

Five different cases have been considered:

**•** Case 2: one TCSC is installed,

**•** Case 3: one SVC is installed,

**Device Type**

**Size/Location**

**Size (MVA)**

**Table 1.** The results for FACTS allocations and sizes

**•** Case 4: one UPFC is installed, and

**Figure 3.** Cost Function of the FACTS devices: SVC, TCSC, and UPFC

#### **4.3. Objective function**

The goal of the optimization algorithm is to place FACTS devices in order to enhance power system security-level considering cost function of FACTS devices. These devices should be placed to prevent congestion in transmission lines and transformers, and to maintain bus voltages close to their reference values. Security index introduced in section III has been used in objective function considering cost function of FACTS devices and power system losses. Fitness function *FF* is expressed by the following equation:

$$FF = a\_1 J\_p + a\_2 J\_v + a\_3 \left( Total\ Investment\ Cost\right) + a\_4 \left(Losses\right) \tag{15}$$

The coefficient *a*1 to *a*<sup>4</sup> have been selected by trial and error, which are 0.2665, 0.5714, 0.1421, and 0.02, respectively. These values *a*1 to *a*<sup>4</sup> give better optimization results for different runs of the algorithm. In the equation above, the third term is sum of cost functions of TCSC, UPFC and SVC, described in equations (12) to (14). In addition, the forth term is total active power loss of the power system.

## **5. Simulation results**

Here, *s* is the operating range of the FACTS devices in MVAR, and *CTCSC*, *CSVC*, and *CUPFC* are

The goal of the optimization algorithm is to place FACTS devices in order to enhance power system security-level considering cost function of FACTS devices. These devices should be placed to prevent congestion in transmission lines and transformers, and to maintain bus voltages close to their reference values. Security index introduced in section III has been used in objective function considering cost function of FACTS devices and power system losses.

The coefficient *a*1 to *a*<sup>4</sup> have been selected by trial and error, which are 0.2665, 0.5714, 0.1421, and 0.02, respectively. These values *a*1 to *a*<sup>4</sup> give better optimization results for different runs of the algorithm. In the equation above, the third term is sum of cost functions of TCSC, UPFC

( ) ( ) <sup>123</sup> <sup>4</sup> . . *p v FF a J a J a Total Investment Cost a Losses* =++ + (15)

in *\$US* / *kVAR*.

208 Search Algorithms for Engineering Optimization

**Figure 3.** Cost Function of the FACTS devices: SVC, TCSC, and UPFC

Fitness function *FF* is expressed by the following equation:

**4.3. Objective function**

Simulation studies are carried out for different scenarios in the IEEE 30-bus power system. Five different cases have been considered:


The first case is the normal operation of network without using any FACTS device. In the second, third, and fourth cases, installation of only one device has been considered. Each device is placed in an optimal location obtained by HSA. Multi-type FACTS devices installation is considered in the 5th scenario. In this case, three different kinds of FACTS devices have been considered to be placed in optimal locations to enhance power system security.

The performance index evolutions of implemented methods are shown in Fig.4, and Fig.5. The average, and maximum performance indices are shown in Fig.4. Tables 1, and 2 show optimal locations of devices for different cases. These results illustrate that the installation of one device in the network could not lead to improved security of power system and reduction in power system losses simultaneously, and that multi-type FACTS devices should be placed in optimal locations to improve security margins and reduce losses in the network.


**Table 1.** The results for FACTS allocations and sizes


**6. Harmonic optimization in multi-level inverters using harmony search**

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211

Nowadays, dc-to-ac inverters are widely used in industry. All applications are mainly divided into two general groups; 1- Electric drives for all ac motors when dc supply is used, and 2- in systems including high voltage direct current (HVDC) transmission systems, custom power and flexible ac transmission systems (FACTS) devices, flexible distributed generation (FDG), and interconnection of distributed generation (DG) units to a grid. Several switching algorithm such as pulse width modulation (PWM), sinusoidal pulse width modulation (SPWM), spacevector modulation (SVM), selective harmonic eliminated pulse width modulation (SHEPWM), or programmed-waveform pulse width modulation (PWPWM) are applied extensively to control and determine switching angles to achieve the desired output voltage. In the recent decade, a new kind of inverter named multi-level inverter has been introduced. In various publications, this inverter has been used in place of the common inverters to indicate its advantages in different applications. Being multi-level, it can be used in high-power and highvoltage applications. In order to reach the desired fundamental component of voltage, all of various switching methods produce harmonics and hence, it is of interest to select the best method to achieve minimum harmonics and total harmonic distortion (THD). It is suggested to use optimized harmonic stepped waveform (OHSW) to eliminate low order harmonics by determining proper angles, and then removing the rest of the harmonics via filters. In addition, this technique lowers switching frequency down to the fundamental frequency and conse‐

Traditionally, there are two states for DC sources in multi-level inverters: 1- Equal DC sources, 2- Non-equal DC sources. Several algorithms have been suggested for the above purposes. In [37] Newton-Raphsonmethodhasbeenusedtosolveequations.Newton-Raphsonmethodisfastand exact for those modulation indices (*M*) that can satisfy equations, but it cannot obtain the best answer for other indices. Also, [38] has used the mathematical theory of resultants to find the switching angles such that all corresponding low-order harmonics are completely canceled out sequentially for both equal and non-equal DC sources separately. However, by increasing levels ofmulti-levelconverters,equationsettendstoahigh-orderpolynomial,whichnarrowsitsfeasible solution space. In addition, this method cannot suggest any answer to minimize harmonics of some particular modulation indices where there is no acceptable solution for the equation set. Genetic algorithm (GA) method has been presented in [39] to solve the same problem with any number of levels for both eliminating and minimizing the harmonics, but it is not fast and exact enough. This method has also been used in [40] to eliminate the mentioned harmonics for nonequal DC sources. Moreover, all optimal solutions have used main equations in fitness function.

Here, a harmony search (HS) algorithm approach will be presented that can solve the problem with a simpler formulation and with any number of levels without extensive derivation of

This means that the fundamental component cannot be satisfied exactly.

analytical expressions. It is also faster and more precise than GA.

**algorithm**

**6.1. Introduction**

quently, power losses and cost are reduced.

**Table 2.** Simulation results for different cases

**Figure 4.** Performance index evolution (average of fitnesses in every iteration)

**Figure 5.** Performance index evolution (maximum of fitnesses in every iteration)

## **6. Harmonic optimization in multi-level inverters using harmony search algorithm**

## **6.1. Introduction**

**Scenario** *JP JV* **(\$) Cost Losses (MVA)** 3.45 24.2 - 28.36 3.39 19.6 9197500 22.51 3.33 19.1 4521600 26.3 3.21 16.4 8501600 23.09 3.09 10.5 27897000 20.49

**Table 2.** Simulation results for different cases

210 Search Algorithms for Engineering Optimization

**Figure 4.** Performance index evolution (average of fitnesses in every iteration)

**Figure 5.** Performance index evolution (maximum of fitnesses in every iteration)

Nowadays, dc-to-ac inverters are widely used in industry. All applications are mainly divided into two general groups; 1- Electric drives for all ac motors when dc supply is used, and 2- in systems including high voltage direct current (HVDC) transmission systems, custom power and flexible ac transmission systems (FACTS) devices, flexible distributed generation (FDG), and interconnection of distributed generation (DG) units to a grid. Several switching algorithm such as pulse width modulation (PWM), sinusoidal pulse width modulation (SPWM), spacevector modulation (SVM), selective harmonic eliminated pulse width modulation (SHEPWM), or programmed-waveform pulse width modulation (PWPWM) are applied extensively to control and determine switching angles to achieve the desired output voltage. In the recent decade, a new kind of inverter named multi-level inverter has been introduced. In various publications, this inverter has been used in place of the common inverters to indicate its advantages in different applications. Being multi-level, it can be used in high-power and highvoltage applications. In order to reach the desired fundamental component of voltage, all of various switching methods produce harmonics and hence, it is of interest to select the best method to achieve minimum harmonics and total harmonic distortion (THD). It is suggested to use optimized harmonic stepped waveform (OHSW) to eliminate low order harmonics by determining proper angles, and then removing the rest of the harmonics via filters. In addition, this technique lowers switching frequency down to the fundamental frequency and conse‐ quently, power losses and cost are reduced.

Traditionally, there are two states for DC sources in multi-level inverters: 1- Equal DC sources, 2- Non-equal DC sources. Several algorithms have been suggested for the above purposes. In [37] Newton-Raphsonmethodhasbeenusedtosolveequations.Newton-Raphsonmethodisfastand exact for those modulation indices (*M*) that can satisfy equations, but it cannot obtain the best answer for other indices. Also, [38] has used the mathematical theory of resultants to find the switching angles such that all corresponding low-order harmonics are completely canceled out sequentially for both equal and non-equal DC sources separately. However, by increasing levels ofmulti-levelconverters,equationsettendstoahigh-orderpolynomial,whichnarrowsitsfeasible solution space. In addition, this method cannot suggest any answer to minimize harmonics of some particular modulation indices where there is no acceptable solution for the equation set. Genetic algorithm (GA) method has been presented in [39] to solve the same problem with any number of levels for both eliminating and minimizing the harmonics, but it is not fast and exact enough. This method has also been used in [40] to eliminate the mentioned harmonics for nonequal DC sources. Moreover, all optimal solutions have used main equations in fitness function. This means that the fundamental component cannot be satisfied exactly.

Here, a harmony search (HS) algorithm approach will be presented that can solve the problem with a simpler formulation and with any number of levels without extensive derivation of analytical expressions. It is also faster and more precise than GA.

## **7. Cascade H-bridges**

The cascaded multi-level inverter is one of the several multi-level configurations. It is formed by connecting several single-phase, H-bridge converters in series as shown in Fig. 1a for a 13 level inverter. Each converter generates a square-wave voltage waveform with different duty ratios. Together, these form the output voltage waveform, as shown in Fig. 1b. A three-phase configuration can be obtained by connecting three of these converters in *Y* , or *Δ*. For harmonic optimization, the switching angles *θ*1, *θ*2,… and *θ*6 (for a 13-level inverter) shown in Fig. 1b have to be selected, so that certain order harmonics are eliminated.

## **8. Problem statement**

Fig. 6b shows a 13-level inverter, where *θ*1, *θ*2,…and *θ*6 are variables and should be determined. Each full-bridge inverter produces a three level waveform +*Vdc*, −*Vdc* and 0, and each angle *θ<sup>i</sup>* is related to the *i* th inverter *i* =1, 2, ..., *S*. *S* is the number of DC sources that is equal to the number of switching angles (in this study S=6). The number of levels *L*, is calculated as *L* =2*S* + 1. Considering equal amplitude of all dc sources, the Fourier series expansion of the output voltage waveform is as follows:

$$V\left(t\right) = \sum\_{n=1}^{\infty} V\_n \sin\left(n\alpha t\right) \tag{16}$$

where, *Vn* is the amplitude of harmonics. The angles are limited between 0 and 90 (0<*θ<sup>i</sup>* <*π* / 2). Because of odd quarter-wave symmetric characteristic, the harmonics with even orders become zero. Consequently, *Vn* will be as follows:

$$V\_n = \begin{cases} \frac{4V\_{dc}}{n\pi} \sum\_{i=1}^{k} \cos\left(n\theta\_i\right) & \text{for odd } n\\ 0 & \text{for even } n \end{cases} \tag{17}$$

Figure 6. a) Multi-Level Inverter b) Multi-Level waveform generation

Chromosome structure of a GA is shown in table 1 that involves *<sup>i</sup>*

In order to optimize the THD, genetic algorithm (GA) that is based on natural evolution and population is implemented. This algorithm is usually applied to reach a near global optimum solution. In each iteration of GA (referred as generation), a new set of strings (.i.e. chromosomes) with improved fitness is produced using genetic operators (i.e. selection, crossover and mutation).

( ) ( ) ( )

qqq

( ) ( ) ( ) ( ) ( ) ( )

qqq

cos cos ... cos 0 cos 5 cos 5 ... cos 5

= + ++ = + ++

0 cos 17 cos 17 ... cos 17

= + ++

12 6 12 6

qqq

b)

 5

> 6

 4

a)

 3 /2 2

*S*1

*S* <sup>2</sup>

*S* <sup>4</sup>

*S* 2

*S* 4

*S* 2

*S* 4

*S* 2

*S* 4

*Vdc*

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*Vdc*

*Vdc*

*Vdc*

*S* 3

*V*1

*Vn* ( 6) *n* *S*1

*S* 3

*S*1

*S* 3

*S*1

*S* 3

 1 2 3

/ 2

1

2

5

4  3

6 

*Vdc*

**Figure 6.** a) Multi-Level Inverter b) Multi-Level waveform generation

*M*

*Vdc*

*Vdc*

*Vdc*

*Vdc*

*Vdc*

6 *Vdc*

6*Vdc*

0

4.75 13.02 30.26 43.55 87.36 89.82

12 6

The method of tournament selection is used for selections in a GA [41-42]. This method chooses each parent by choosing *<sup>t</sup> n* (Tournament size) players randomly, and choosing the best individual out of that set to be a parent. In this section, *<sup>t</sup> n* is chosen as

Crossover allows the genes from different parents to be combined in children by exchanging materials between two parents. Crossover function randomly selects a gene at the same coordinate from one of the two parents and assigns it to the child. For each chromosome, a random number is selected. If this number is between 0.01 and 0.3 [42], the two parents are combined; else

as parameter of the inverter.

<sup>M</sup> (18)

**9. Genetic algorithm** 

Table 3. A typical chromosome

a. Chromosome's structure

b. Selection

c. Cross Over

chromosome is transferred with no crossover.

4.

There are two approaches to adjust the switching angles:


The second approach is preferred. For motor drive applications, it is necessary to eliminate low order harmonics from 5 to 17. Hence, in this section, a 13-level inverter is chosen to eliminate low-order harmonics from 5 to 17. It is not needed to delete triple harmonics because they will be eliminated in three-phase circuits. Thus, for a 13-level inverter, Eq. (17) changes into (18).

Application of Harmony Search Algorithm in Power Engineering http://dx.doi.org/10.5772/55509 213

Figure 6. a) Multi-Level Inverter b) Multi-Level waveform generation **Figure 6.** a) Multi-Level Inverter b) Multi-Level waveform generation

Chromosome structure of a GA is shown in table 1 that involves *<sup>i</sup>*

**9. Genetic algorithm** 

Table 3. A typical chromosome

a. Chromosome's structure

b. Selection

c. Cross Over

chromosome is transferred with no crossover.

4.

**7. Cascade H-bridges**

212 Search Algorithms for Engineering Optimization

**8. Problem statement**

output voltage waveform is as follows:

is related to the *i*

into (18).

The cascaded multi-level inverter is one of the several multi-level configurations. It is formed by connecting several single-phase, H-bridge converters in series as shown in Fig. 1a for a 13 level inverter. Each converter generates a square-wave voltage waveform with different duty ratios. Together, these form the output voltage waveform, as shown in Fig. 1b. A three-phase configuration can be obtained by connecting three of these converters in *Y* , or *Δ*. For harmonic optimization, the switching angles *θ*1, *θ*2,… and *θ*6 (for a 13-level inverter) shown in Fig. 1b

Fig. 6b shows a 13-level inverter, where *θ*1, *θ*2,…and *θ*6 are variables and should be determined. Each full-bridge inverter produces a three level waveform +*Vdc*, −*Vdc* and 0, and each angle *θ<sup>i</sup>*

number of switching angles (in this study S=6). The number of levels *L*, is calculated as *L* =2*S* + 1. Considering equal amplitude of all dc sources, the Fourier series expansion of the

> ( ) ( ) 1

*n Vt V nt*

1

*k dc*

cos

*<sup>V</sup> n for oddn <sup>V</sup> <sup>n</sup>*

orders become zero. Consequently, *Vn* will be as follows:

There are two approaches to adjust the switching angles:

4

æ ç = ç ç è

0

*n i*

p= =

¥

sin *<sup>n</sup>*

where, *Vn* is the amplitude of harmonics. The angles are limited between 0 and 90 (0<*θ<sup>i</sup>* <*π* / 2). Because of odd quarter-wave symmetric characteristic, the harmonics with even

( )

**1.** Minimizing the THD that is not common, because some low order harmonics may remain. **2.** Canceling the lower order harmonics and removing the remained harmonics with a filter. The second approach is preferred. For motor drive applications, it is necessary to eliminate low order harmonics from 5 to 17. Hence, in this section, a 13-level inverter is chosen to eliminate low-order harmonics from 5 to 17. It is not needed to delete triple harmonics because they will be eliminated in three-phase circuits. Thus, for a 13-level inverter, Eq. (17) changes

q

*i*

*for even n*

w

th inverter *i* =1, 2, ..., *S*. *S* is the number of DC sources that is equal to the

<sup>=</sup> å (16)

å (17)

have to be selected, so that certain order harmonics are eliminated.

$$\begin{aligned} M &= \cos\left(\theta\_1\right) + \cos\left(\theta\_2\right) + ... + \cos\left(\theta\_b\right) \\ 0 &= \cos\left(5\theta\_1\right) + \cos\left(5\theta\_2\right) + ... + \cos\left(5\theta\_b\right) \\ \vdots \\ 0 &= \cos\left(17\theta\_1\right) + \cos\left(17\theta\_2\right) + ... + \cos\left(17\theta\_b\right) \end{aligned} \tag{18}$$

as parameter of the inverter.

4.75 13.02 30.26 43.55 87.36 89.82

The method of tournament selection is used for selections in a GA [41-42]. This method chooses each parent by choosing *<sup>t</sup> n* (Tournament size) players randomly, and choosing the best individual out of that set to be a parent. In this section, *<sup>t</sup> n* is chosen as

Crossover allows the genes from different parents to be combined in children by exchanging materials between two parents. Crossover function randomly selects a gene at the same coordinate from one of the two parents and assigns it to the child. For each chromosome, a random number is selected. If this number is between 0.01 and 0.3 [42], the two parents are combined; else Here, *M* is the modulation index and defined as:

$$M = \frac{V\_1}{4V\_{dc}/\pi} \quad \left(0 < M \le 6\right) \tag{19}$$

**10. Harmony search algorithm**

(20)

1 2 1 <sup>1</sup> <sup>100</sup> <sup>50</sup> *<sup>i</sup> i V V <sup>V</sup> <sup>f</sup> <sup>V</sup> h V* 

**10. Harmony search algorithm** 

**11. Simulation Results** 

<sup>4</sup> \* <sup>6</sup> 1 1 \*

d. Mutation

chapter [34-35].

in section 4 of the first part of this chapter [34-35].

search algorithm performed much better than GA approach.

Figure 7. a) output voltage waveform b) harmonic spectrum

function has been chosen as follows:

**11. Simulation results**

**Figure 7.** a) output voltage waveform b) harmonic spectrum

0.1 [42], mutation process is applied; else chromosome is transferred with no mutation.

Harmony Search Algorithm (HSA) has been implemented based on the algorithm described

a)

b)

Harmony Search algorithm has been used to solve the optimization problem. The objective

1 2 1

å (20)

<sup>4</sup> \* <sup>6</sup> 1 1 \*

*V V <sup>V</sup> <sup>f</sup> <sup>V</sup>* <sup>=</sup> *h V* ìü ïï æö - æö =ç÷+ íý ç÷ ç÷ ïèø èøï îþ

<sup>1</sup> <sup>100</sup> <sup>50</sup> *<sup>i</sup> i*

where, \* *<sup>V</sup>*1 is the desired fundamental harmonic, *h1=1* , *h2=5* … and *h6=*17, are orders of the first six viable harmonics at the output of a three-phase multi-level inverter, respectively. The parameters of the harmony search algorithm have been chosen as: *HMS*=10, *HMCR*=0.9, *PAR*=0.6, and *bw* =0.01. The optimal solution vector is obtained after 1000 iterations as: [10.757, 16.35, 26.973, 39.068, 59.409, 59.409]. With these switching angles, the output voltage waveform and its spectrum will be obtained as shown in Fig. 2. The values of the objective function and the total harmonic distortion (THD) has been obtained as: *THD* 4.73% , and *f* 4.8 8 *e* . Simulation has also been performed by GA and results obtained as: *THD* 7.11% , and *f* 0.05 . It is obvious that the harmony

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GA creates mutation-children by randomly changing the genes of individual parents. In this section, GA adds a random vector from a Gaussian distribution to the parents. For each chromosome, random number is selected. If this number is between 0.01 and

Harmony Search Algorithm (HSA) has been implemented based on the algorithm described in section 4 of the first part of this

Harmony Search algorithm has been used to solve the optimization problem. The objective function has been chosen as follows:

It is necessary to determine six switching angles, namely *θ*1, *θ*2,…and *θ*6 so that equation set (18) is satisfied. These equations are nonlinear and different methods can be applied to solve them.

## **9. Genetic algorithm**

In order to optimize the THD, genetic algorithm (GA) that is based on natural evolution and population is implemented. This algorithm is usually applied to reach a near global optimum solution. In each iteration of GA (referred as generation), a new set of strings (i.e. chromosomes) withimprovedfitnessisproducedusinggeneticoperators(i.e.selection,crossoverandmutation).


**Table 3.** A typical chromosome

#### **a.** Chromosome's structure

Chromosome structure of a GA is shown in table 1 that involves *θ<sup>i</sup>* as parameter of the inverter.

**b.** Selection

The method of tournament selection is used for selections in a GA [41-42]. This method chooses each parent by choosing *nt* (Tournament size) players randomly, and choosing the best individual out of that set to be a parent. In this section, *nt* is chosen as 4.

#### **c.** Cross Over

Crossover allows the genes from different parents to be combined in children by exchanging materials between two parents. Crossover function randomly selects a gene at the same coordinate from one of the two parents and assigns it to the child. For each chromosome, a random number is selected. If this number is between 0.01 and 0.3 [42], the two parents are combined; else chromosome is transferred with no crossover.

**d.** Mutation

GA creates mutation-children by randomly changing the genes of individual parents. In this section, GA adds a random vector from a Gaussian distribution to the parents. For each chromosome, random number is selected. If this number is between 0.01 and 0.1 [42], mutation process is applied; else chromosome is transferred with no mutation.

#### **10. Harmony search algorithm** where, \* *<sup>V</sup>*1 is the desired fundamental harmonic, *h1=1* , *h2=5* … and *h6=*17, are orders of the first six viable harmonics at the output

(20)

search algorithm performed much better than GA approach.

1 2 1 <sup>1</sup> <sup>100</sup> <sup>50</sup> *<sup>i</sup> i V V <sup>V</sup> <sup>f</sup> <sup>V</sup> h V* 

**10. Harmony search algorithm** 

**11. Simulation Results** 

<sup>4</sup> \* <sup>6</sup> 1 1 \*

d. Mutation

chapter [34-35].

Here, *M* is the modulation index and defined as:

214 Search Algorithms for Engineering Optimization

**9. Genetic algorithm**

**Table 3.** A typical chromosome

**b.** Selection

**c.** Cross Over

**d.** Mutation

**a.** Chromosome's structure

( ) <sup>1</sup> 0 6

It is necessary to determine six switching angles, namely *θ*1, *θ*2,…and *θ*6 so that equation set (18) is satisfied. These equations are nonlinear and different methods can be applied to solve them.

In order to optimize the THD, genetic algorithm (GA) that is based on natural evolution and population is implemented. This algorithm is usually applied to reach a near global optimum solution. In each iteration of GA (referred as generation), a new set of strings (i.e. chromosomes) withimprovedfitnessisproducedusinggeneticoperators(i.e.selection,crossoverandmutation).

4.75 13.02 30.26 43.55 87.36 89.82

The method of tournament selection is used for selections in a GA [41-42]. This method chooses each parent by choosing *nt* (Tournament size) players randomly, and choosing the best

Crossover allows the genes from different parents to be combined in children by exchanging materials between two parents. Crossover function randomly selects a gene at the same coordinate from one of the two parents and assigns it to the child. For each chromosome, a random number is selected. If this number is between 0.01 and 0.3 [42], the two parents are

GA creates mutation-children by randomly changing the genes of individual parents. In this section, GA adds a random vector from a Gaussian distribution to the parents. For each chromosome, random number is selected. If this number is between 0.01 and 0.1 [42], mutation

= <£ (19)

as parameter of the inverter.

4 *dc <sup>V</sup> M M V* p

Chromosome structure of a GA is shown in table 1 that involves *θ<sup>i</sup>*

combined; else chromosome is transferred with no crossover.

process is applied; else chromosome is transferred with no mutation.

individual out of that set to be a parent. In this section, *nt* is chosen as 4.

Harmony Search Algorithm (HSA) has been implemented based on the algorithm described in section 4 of the first part of this chapter [34-35]. *HMCR*=0.9, *PAR*=0.6, and *bw* =0.01. The optimal solution vector is obtained after 1000 iterations as: [10.757, 16.35, 26.973, 39.068, 59.409, 59.409]. With these switching angles, the output voltage waveform and its spectrum will be obtained as shown in Fig. 2. The values of the objective function and the total harmonic distortion (THD) has been obtained as: *THD* 4.73% , and *f* 4.8 8 *e* . Simulation has also been performed by GA and results obtained as: *THD* 7.11% , and *f* 0.05 . It is obvious that the harmony

of a three-phase multi-level inverter, respectively. The parameters of the harmony search algorithm have been chosen as: *HMS*=10,

GA creates mutation-children by randomly changing the genes of individual parents. In this section, GA adds a random vector from a Gaussian distribution to the parents. For each chromosome, random number is selected. If this number is between 0.01 and

Harmony Search Algorithm (HSA) has been implemented based on the algorithm described in section 4 of the first part of this

Harmony Search algorithm has been used to solve the optimization problem. The objective function has been chosen as follows:

0.1 [42], mutation process is applied; else chromosome is transferred with no mutation.

Figure 7. a) output voltage waveform b) harmonic spectrum **Figure 7.** a) output voltage waveform b) harmonic spectrum

## **11. Simulation results**

Harmony Search algorithm has been used to solve the optimization problem. The objective function has been chosen as follows:

$$f = \left| \left( 100 \frac{V\_1^\* - V\_1}{V\_1^\*} \right)^4 + \sum\_{i=2}^6 \frac{1}{h} \left( 50 \frac{V\_i}{V\_1} \right) \right| \tag{20}$$

where, *V*<sup>1</sup> \* is the desired fundamental harmonic, *h1=1, h2=5* … and *h6=*17, are orders of the first six viable harmonics at the output of a three-phase multi-level inverter, respectively. The parame‐ ters of the harmony search algorithm have been chosen as: *HMS*=10, *HMCR*=0.9, *PAR*=0.6, and *bw* =0.01.Theoptimalsolutionvectorisobtainedafter1000iterationsas:[10.757,16.35,26.973,39.068, 59.409, 59.409]. With these switching angles, the output voltage waveform and its spectrum will beobtainedasshowninFig.2.Thevaluesoftheobjectivefunctionandthetotalharmonicdistortion (THD) has been obtained as: *THD* =4.73*%*, and *f* =4.8*e* −8. Simulation has also been performed by GA and results obtained as: *THD* =7.11*%*, and *f* =0.05. It is obvious that the harmony search algorithm performed much better than GA approach.

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[8] Kazemi, A, Shayanfar, H. A, Rabiee, A, & Aghaie, J. Power System Security Improve‐ ment using the Unified Power Flow Controller (UPFC)", IEEE Power India Confer‐

[9] Singh, J. G, Singh, S. N, & Srivastava, S. C. Placement of FACTS Controllers for En‐ hancing Power System Loadability" IEEE Power India Conference, (2006). , 1, 1-7.

[10] Jurado, F, & Rodriguez, J. A. Optimal location of SVC based on system loadability and contingency analysis", Proceeding of Emerging Technologies and Automation,

[11] Sing, S. N, & David, A. K. A New Approach for Placement of FACTS Devices in Open Power Markets", IEEE Power Engineering Review, (2001). , 21, 58-60.

[12] Galiana, F. D, Almeida, K, Toussaint, M, Griffin, J, & Atanackovic, D. Assessment and Control of the Impact of FACTS Devices on Power System Performance", IEEE

ogy of Flexible AC Transmission Systems", USA, IEEE Press, New York.

## **12. Conclusion**

In the first part of the presented chapter, we presented a novel approach for optimal placement of multi-type FACTS devices based on harmony search algorithm. Simulations of IEEE 30-bus test system for different scenarios demonstrate that the placement of multi-type FACTS devices leads to improvement in security, and reduction in losses of power systems.

In the second part, the harmony search algorithm was proposed for harmonic optimization in multi-level inverters. Harmony search algorithm has more flexibility than conventional methods. This method can obtain optimum switching angles for a wide range of modulation indices. This advantage is of importance, especially when the number of switching angles goes up, where equation set may not have any solution, or when it is solvable only for a short range of modulation indices. Moreover, the implementation of the harmony search algorithm is very straightforward compared to the conventional methods like Newton-Raphson, where it is necessary to calculate the Jacobean matrix. In addition, one of the most attractive features of intelligent algorithms is their independency from case studies. Actually, intelligent algorithm can be imposed to a variety of different problems without any need for extensive manipula‐ tions. For example, the harmony search algorithm and GA algorithms are able to find optimum switching angles in order to cancel out low-order harmonics, and if it is not possible to completely remove them, they can suggest optimum switching angles so that, low-order harmonics will be reduced as much as possible. Furthermore, with a little manipulation in the defined objective function, one can use HSA and GA as a tool for THD optimization. Also, the results indicate that, harmony search algorithm has many benefits over GA such as simplicity in the implementation, precision, and speed in global convergence.

## **Author details**

H. R. Baghaee\* , M. Mirsalim and G. B. Gharehpetian

\*Address all correspondence to: hrbaghaee@aut.ac.ir

Department of Electrical Engineering, Amirkabir University of Technology, Tehran, Iran

## **References**

where, *V*<sup>1</sup>

\*

216 Search Algorithms for Engineering Optimization

**12. Conclusion**

**Author details**

H. R. Baghaee\*

algorithm performed much better than GA approach.

is the desired fundamental harmonic, *h1=1, h2=5* … and *h6=*17, are orders of the first six

viable harmonics at the output of a three-phase multi-level inverter, respectively. The parame‐ ters of the harmony search algorithm have been chosen as: *HMS*=10, *HMCR*=0.9, *PAR*=0.6, and *bw* =0.01.Theoptimalsolutionvectorisobtainedafter1000iterationsas:[10.757,16.35,26.973,39.068, 59.409, 59.409]. With these switching angles, the output voltage waveform and its spectrum will beobtainedasshowninFig.2.Thevaluesoftheobjectivefunctionandthetotalharmonicdistortion (THD) has been obtained as: *THD* =4.73*%*, and *f* =4.8*e* −8. Simulation has also been performed by GA and results obtained as: *THD* =7.11*%*, and *f* =0.05. It is obvious that the harmony search

In the first part of the presented chapter, we presented a novel approach for optimal placement of multi-type FACTS devices based on harmony search algorithm. Simulations of IEEE 30-bus test system for different scenarios demonstrate that the placement of multi-type FACTS devices

In the second part, the harmony search algorithm was proposed for harmonic optimization in multi-level inverters. Harmony search algorithm has more flexibility than conventional methods. This method can obtain optimum switching angles for a wide range of modulation indices. This advantage is of importance, especially when the number of switching angles goes up, where equation set may not have any solution, or when it is solvable only for a short range of modulation indices. Moreover, the implementation of the harmony search algorithm is very straightforward compared to the conventional methods like Newton-Raphson, where it is necessary to calculate the Jacobean matrix. In addition, one of the most attractive features of intelligent algorithms is their independency from case studies. Actually, intelligent algorithm can be imposed to a variety of different problems without any need for extensive manipula‐ tions. For example, the harmony search algorithm and GA algorithms are able to find optimum switching angles in order to cancel out low-order harmonics, and if it is not possible to completely remove them, they can suggest optimum switching angles so that, low-order harmonics will be reduced as much as possible. Furthermore, with a little manipulation in the defined objective function, one can use HSA and GA as a tool for THD optimization. Also, the results indicate that, harmony search algorithm has many benefits over GA such as simplicity

leads to improvement in security, and reduction in losses of power systems.

in the implementation, precision, and speed in global convergence.

, M. Mirsalim and G. B. Gharehpetian

Department of Electrical Engineering, Amirkabir University of Technology, Tehran, Iran

\*Address all correspondence to: hrbaghaee@aut.ac.ir


[13] Thukaram, D, Jenkins, L, & Visakha, K. Improvement Of System Security With Uni‐ fied-Power-Flow Controller At Suitable Locations Under Network Contingencies Of Interconnected Systems", IEE Proc. Generation, Transmission and Distribution, (2005). , 152, 682-690.

[26] Lie, T. T, & Deng, W. Optimal flexible AC transmission systems (FACTS) devices al‐

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[27] Baskaran, J, & Palanisamy, V. Optimal location of FACTS device in a power system network considering power loss using genetic algorithm", EE-Pub on line journal

[28] Baghaee, H. R, Mirsalim, M, & Kashefi, A. Kaviani, G.B. Gharehpetian, "Security/ cost-based optimal allocation of multi-type FACTS devices using multi-objective par‐ ticle swarm optimization", Simulation: Transactions of the Society for Modeling and

[29] Kim, S. H, Lim, J. U, & Moon, S. I. I. Enhancement of power system security level through the power control of UPFC", IEEE Power Engineering Society Summer

[30] Kim, T. H, Ch, J, Seo, J. U, Lim, S, Moon, J. K, & Park, B. M. Han," A Decoupled uni‐ fied power flow controller model for power flow considering limit resolution", IEEE

[31] Lim, J. U, & Moon, S. I. An operation scheme of UPFC's considering operation objec‐ tive and states", IEEE Power Engineering Society Winter Meeting, (2002). , 1, 610-615.

[32] Atif, S. Debs, " Modem Power Systems Control and Operation" USA, (2008). Kluwer

[33] Allen, J. Wood, Bruce F. Wollenberg, "Power Generation, Operation, and Control",

[34] Alireza Askarzadeh, Alireza Rezazadeh "Parameter identification for solar cell mod‐ els using harmony search-based algorithms", Solar Energy, (2012). , 86, 3241-3249. [35] Sergio Gil-López, Javier Del Ser, Sancho Salcedo-Sanz, Ángel M. Pérez-Bellido, José Marı´a Cabero, José A. Portilla-Figueras "A hybrid harmony search algorithm for the spread spectrum radar polyphase codes design problem", Expert Systems with Ap‐

[36] Lee, K. S, & Geem, Z. W. A New Structural Optimization Method Based On The Har‐

[37] Jason LaiChair Dusan Borojevic, and Alex Q. Huang "Optimized Harmonic Stepped Waveform from Multi-Level Inverters", M.S thesis Virginia Polytechnic Institute and

[38] Chiasson, J, & Tolbert, L. Keith McKenzie, and Zhong Du, "Eliminating Harmonics in a Multi-level Converter using Resultant Theory", IEEE Trans. on Control Systems

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[13] Thukaram, D, Jenkins, L, & Visakha, K. Improvement Of System Security With Uni‐ fied-Power-Flow Controller At Suitable Locations Under Network Contingencies Of Interconnected Systems", IEE Proc. Generation, Transmission and Distribution,

[14] Verma, K. S, Singh, S. N, & Gupta, H. O. Location of Unified Power Flow Controller for Congestion Management", Electric Power System Research, , 58, 89-96.

[15] Gerbex, S, Chekaoui, R, & Germond, A. J. Optimal Location of Multi-type FACTS De‐ vices in a Power System be Means of Genetic Algorithm", IEEE Trans. on Power Sys‐

[16] Gerbex, S, Cherkaoui, R, & Germond, A. J. Optimal Location of FACTS Devices to Enhance Power System Security", IEEE Power Tech Conference Proceedings, (2003).

[17] Cai, L. J. and Erlich, "Optimal choice and allocation of FACTS devices in deregulated electricity market using genetic algorithms", IEEE PES Power Systems Conference

[18] Baskaran, J, & Palanisamy, V. Genetic Algorithm Applied to Optimal Location of FACTS Device in a Power System Network Considering Economic Saving Cost",

[19] Chung-Fu Chang, Ji-Jen Wong, Ji-Pyng Chiouand Ching-Tzong Su "Robust Search‐ ing Hybrid Differential Evolution Method for Optimal Reactive Power Planning in Large-Scale Distribution Systems", Electric Power System Research, (2007). , 77,

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[22] Gerbex, R, & Cherkaoui, A. J. Germond "Optimal Location Of Multi-Type FACTS Devices In A Power System By Means Of Genetic Algorithm", IEEE Trans. On Power

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[39] Ozpineci, B, Tolbert, L. M, & Chiasson, J. N. Harmonic Optimization of Multi-level Converters using Genetic Algorithms", Proceeding on 35th IEEE Power Electronics Specialists Conference, ((2005). , 5, 3911-3916.

**Chapter 9**

**Provisional chapter**

**Heuristic Search Applied to Fuzzy Cognitive Maps**

**Heuristic Search Applied to Fuzzy Cognitive Maps**

Fuzzy Cognitive Maps were initially proposed by Kosko [1–3], as an extension of cognitive maps proposed by Axelrod [4]. FCM is a graph used for representing causal relationships among concepts that stand for the states and variables of the system, emulating the cognitive knowledge of experts on a specific area. FCM can be interpreted as a combination of Fuzzy Logic and Neural Networks, because it combines the sense rules of Fuzzy Logic with the learning of the Neural Networks. A FCM describes the behavior of a knowledge based system in terms of concepts, where each concept represents an entity, a state, a variable, or a characteristic of the system. The human knowledge and experience about the system determines the type and the number of the nodes as well as the initial conditions of the FCM. FCM has been considered with great research interest in many scientific fields, such as political decision [5], medical decision [6, 7], industrial process control [8, 9], artificial life [10],

The knowledge of the experts firstly defines the influence of a concept on the other, determining the causality relationships. Then, the concept values are qualitatively obtained by linguistic terms, such as strong, weak, null, and so on. These linguistic variables are transformed in numerical values using a *defuzzification* method, for instance, the center of

Hence, in general, experts develop a FCM identifying key concepts, defining the causal relationships among the concepts, and estimating the strength of these relationships. However, when the experts are not able to express the causal relationships or they substantially diverge in opinion about it, data driven methods for learning FCMs may be

> ©2012 Angélico et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Angélico et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2013 Angélico et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

social systems [11], corporative decision, policy analysis [12], among others.

Bruno Augusto Angélico, Márcio Mendonça, Lúcia Valéria R. de Arruda and Taufik Abrão

Bruno Augusto Angélico, Márcio Mendonça, Lúcia Valéria R. de Arruda, Taufik Abrão

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52918

gravity scheme described in [2].

necessary.

**1. Introduction**

**Learning**

**Learning**


**Provisional chapter**

## **Heuristic Search Applied to Fuzzy Cognitive Maps Learning Heuristic Search Applied to Fuzzy Cognitive Maps Learning**

Bruno Augusto Angélico, Márcio Mendonça, Lúcia Valéria R. de Arruda and Taufik Abrão Lúcia Valéria R. de Arruda, Taufik Abrão Additional information is available at the end of the chapter

Bruno Augusto Angélico, Márcio Mendonça,

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52918

## **1. Introduction**

[39] Ozpineci, B, Tolbert, L. M, & Chiasson, J. N. Harmonic Optimization of Multi-level Converters using Genetic Algorithms", Proceeding on 35th IEEE Power Electronics

[40] Hosseini, M. G, & Aghdam, S. H. Fathi, and A. Ghasemi, "Modeling and Simulation of Three-Phase OHSW Multi-level Voltage-Source Inverter by Means of Switching Functions", IEEE International Conference on Power Electronics and Drives Systems,

[41] Miller, B. L, & Goldberg, D. E. Genetic Algorithms, Tournament Selection, and the

[42] The Math Works IncMATLAB Genetic Algorithm and Direct Search Toolbox User's

Specialists Conference, ((2005). , 5, 3911-3916.

Effects of Noise", Complex Systems, (1995). , 1, 193-212.

(2005). , 1, 633-641.

220 Search Algorithms for Engineering Optimization

Guide, Mathwork Inc., (2012).

Fuzzy Cognitive Maps were initially proposed by Kosko [1–3], as an extension of cognitive maps proposed by Axelrod [4]. FCM is a graph used for representing causal relationships among concepts that stand for the states and variables of the system, emulating the cognitive knowledge of experts on a specific area. FCM can be interpreted as a combination of Fuzzy Logic and Neural Networks, because it combines the sense rules of Fuzzy Logic with the learning of the Neural Networks. A FCM describes the behavior of a knowledge based system in terms of concepts, where each concept represents an entity, a state, a variable, or a characteristic of the system. The human knowledge and experience about the system determines the type and the number of the nodes as well as the initial conditions of the FCM.

FCM has been considered with great research interest in many scientific fields, such as political decision [5], medical decision [6, 7], industrial process control [8, 9], artificial life [10], social systems [11], corporative decision, policy analysis [12], among others.

The knowledge of the experts firstly defines the influence of a concept on the other, determining the causality relationships. Then, the concept values are qualitatively obtained by linguistic terms, such as strong, weak, null, and so on. These linguistic variables are transformed in numerical values using a *defuzzification* method, for instance, the center of gravity scheme described in [2].

Hence, in general, experts develop a FCM identifying key concepts, defining the causal relationships among the concepts, and estimating the strength of these relationships. However, when the experts are not able to express the causal relationships or they substantially diverge in opinion about it, data driven methods for learning FCMs may be necessary.

Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Angélico et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Angélico et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

©2012 Angélico et al., licensee InTech. This is an open access chapter distributed under the terms of the

Particularly, this Chapter focuses on the FCM learning using three different population based metaheuristics: particle swarm optimization (PSO), genetic algorithm (GA) and differential evolution (DE). Two process control problems described in [8] and [9] are considered in this work. A complete convergence analysis of the PSO, GA and DE is carried out considering 10000 realizations of each algorithm in every scenario of the studied processes.

**2.1. First control system (PROC1)**

measures the specific gravity of the resultant mixture.

**Figure 1.** PROC1: a chemical process control problem described in [8].

controller has to keep *G* and *H* within their bounds, i.e.,

• Concept *C*2: state of *V*<sup>1</sup> (closed, open or partially open); • Concept *C*3: state of *V*<sup>2</sup> (closed, open or partially open); • Concept *C*4: state of *V*<sup>3</sup> (closed, open or partially open); • Concept *C*5: specific gravity of the produced mixture.

physical quantities of the process [8]:

A simple chemical process frequently considered in literature [8, 13, 14], is initially selected for illustrating the need of a FCM learning technique. Figure 1 represents the process (PROC1) consisting of one tank and three valves that control the liquid level in the tank. Valves *V*<sup>1</sup> and *V*<sup>2</sup> fill the tank with different liquids. A chemical reaction takes place into the tank producing a new liquid that leaves the recipient by valve *V*3. A sensor (gauger)

When the value of the specific gravity, *G*, is in the range [*G*min, *G*max], the desired liquid has been produced. The height of the liquid inside, *H*, must lie in the range [*H*min, *H*max]. The

The group of experts defined a list of five concepts, *Ci*, *i* = 1, 2, . . . , 5, related to the main

• Concept *C*1: volume of liquid inside the tank (depends on *V*1, *V*2, and, *V*3);

For this process, the fuzzy cognitive map in Figure 2 can be abstracted [8].

Heuristic Search Applied to Fuzzy Cognitive Maps Learning

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223

*H*min ≤ *H* ≤ *H*max; (3) *G*min ≤ *G* ≤ *G*max. (4)

The rest of the Chapter has the following organization: Section 2 briefly describes the FCM modeling and the processes to be controlled. Section 3 considers the PSO, GA and the DE approaching for FCM learning, while Section 4 shows the simulation results. Lastly, Section 5 points out the main conclusions.

## **2. FCM modeling in control processes**

In FCMs, concepts (nodes) are utilized to represent different aspects and behavior of the system. The system dynamics are simulated by the interaction of concepts. The concept *Ci*, *i* = 1, 2, . . . , *N* is characterized by a value *Ai* ∈ [0, 1].

Concepts are interconnected concerning the underlying causal relationships amongst factors, characteristics, and components that constitute the system. Each interconnection between two concepts, *Ci* and *Cj*, has a weight, *Wi*,*j*, which is numerically represented by the strength of the causal relationships between *Ci* and *Cj*. The sign of *Wi*,*<sup>j</sup>* indicates whether the concept *Ci* causes the concept *Cj* or vice versa. Hence, if:

> *Wi*,*<sup>j</sup>* > 0, positive causality *Wi*,*<sup>j</sup>* < 0, negative causality *Wi*,*<sup>j</sup>* = 0, no relation

The number of concepts and the initial weights of the FCM are determined by human knowledge and experience. The numerical values, *Ai*, of each concept is a transformation of the fuzzy values assigned by the experts. The FCM converges to a steady state (limit cycle) according to the scheme proposed in [3]:

$$A\_{\dot{l}}\left(k+1\right) = f\left(A\_{\dot{l}}\left(k\right) + \sum\_{\substack{j=1\\j\neq i}}^{N} \mathcal{W}\_{\dot{l}\dot{l}} A\_{\dot{j}}\left(k\right)\right),\tag{1}$$

where *k* is the interaction index and *f*(·) is the sigmoid function

$$f(\mathbf{x}) = \frac{1}{1 + e^{-\lambda \mathbf{x}'}} \tag{2}$$

that guarantees the values *Ai* ∈ [0, 1]. *λ* > 0 is a parameter representing the learning memory. In this work, *λ* = 1 has been adopted.

## **2.1. First control system (PROC1)**

2 Search Algorithms

5 points out the main conclusions.

**2. FCM modeling in control processes**

*Ci* causes the concept *Cj* or vice versa. Hence, if:

according to the scheme proposed in [3]:

memory. In this work, *λ* = 1 has been adopted.

 

*Ai* (*k* + 1) = *f*

where *k* is the interaction index and *f*(·) is the sigmoid function

*i* = 1, 2, . . . , *N* is characterized by a value *Ai* ∈ [0, 1].

Particularly, this Chapter focuses on the FCM learning using three different population based metaheuristics: particle swarm optimization (PSO), genetic algorithm (GA) and differential evolution (DE). Two process control problems described in [8] and [9] are considered in this work. A complete convergence analysis of the PSO, GA and DE is carried out considering

The rest of the Chapter has the following organization: Section 2 briefly describes the FCM modeling and the processes to be controlled. Section 3 considers the PSO, GA and the DE approaching for FCM learning, while Section 4 shows the simulation results. Lastly, Section

In FCMs, concepts (nodes) are utilized to represent different aspects and behavior of the system. The system dynamics are simulated by the interaction of concepts. The concept *Ci*,

Concepts are interconnected concerning the underlying causal relationships amongst factors, characteristics, and components that constitute the system. Each interconnection between two concepts, *Ci* and *Cj*, has a weight, *Wi*,*j*, which is numerically represented by the strength of the causal relationships between *Ci* and *Cj*. The sign of *Wi*,*<sup>j</sup>* indicates whether the concept

> *Wi*,*<sup>j</sup>* > 0, positive causality *Wi*,*<sup>j</sup>* < 0, negative causality *Wi*,*<sup>j</sup>* = 0, no relation

The number of concepts and the initial weights of the FCM are determined by human knowledge and experience. The numerical values, *Ai*, of each concept is a transformation of the fuzzy values assigned by the experts. The FCM converges to a steady state (limit cycle)

*Ai* (*k*) +

*<sup>f</sup>*(*x*) = <sup>1</sup>

that guarantees the values *Ai* ∈ [0, 1]. *λ* > 0 is a parameter representing the learning

*N* ∑ *j*=1 *j*�=*i*

*WjiAj* (*k*)

<sup>1</sup> <sup>+</sup> *<sup>e</sup>*−*λ<sup>x</sup>* , (2)

, (1)

10000 realizations of each algorithm in every scenario of the studied processes.

A simple chemical process frequently considered in literature [8, 13, 14], is initially selected for illustrating the need of a FCM learning technique. Figure 1 represents the process (PROC1) consisting of one tank and three valves that control the liquid level in the tank. Valves *V*<sup>1</sup> and *V*<sup>2</sup> fill the tank with different liquids. A chemical reaction takes place into the tank producing a new liquid that leaves the recipient by valve *V*3. A sensor (gauger) measures the specific gravity of the resultant mixture.

**Figure 1.** PROC1: a chemical process control problem described in [8].

When the value of the specific gravity, *G*, is in the range [*G*min, *G*max], the desired liquid has been produced. The height of the liquid inside, *H*, must lie in the range [*H*min, *H*max]. The controller has to keep *G* and *H* within their bounds, i.e.,

$$H\_{\rm min} \le H \le H\_{\rm max} \; \text{\;} \tag{3}$$

$$
\mathcal{G}\_{\rm min} \le \mathcal{G} \le \mathcal{G}\_{\rm max}.\tag{4}
$$

The group of experts defined a list of five concepts, *Ci*, *i* = 1, 2, . . . , 5, related to the main physical quantities of the process [8]:


For this process, the fuzzy cognitive map in Figure 2 can be abstracted [8].

**Figure 2.** Fuzzy Cognitive Map proposed in [8] for the chemical process control problem.

The experts also had a consensus regarding the range of the weights between concepts, as presented in Equations (5a) to (5h).

$$-0.50 \le W\_{1,2} \le -0.30;\tag{5a}$$

**2.2. Second control system (PROC2)**

second (PROC2), as illustrated in Figure 3.

**Figure 3.** PROC2: a chemical process control problem described in [9].

*T*max, such that

into tank 2.

In [9] it is considered a system consisting of two identical tanks with one input and one output valve each one, with the output valve of the first tank being the input valve of the

The objective is to control the volume of liquid within the limits determined by the height *H*min and *H*max and the temperature of the liquid in both tanks within the limits *T*min and

min <sup>≤</sup> *<sup>T</sup>*<sup>1</sup> <sup>≤</sup> *<sup>T</sup>*<sup>1</sup>

min <sup>≤</sup> *<sup>T</sup>*<sup>2</sup> <sup>≤</sup> *<sup>T</sup>*<sup>2</sup>

min <sup>≤</sup> *<sup>H</sup>*<sup>1</sup> <sup>≤</sup> *<sup>H</sup>*<sup>1</sup>

min <sup>≤</sup> *<sup>H</sup>*<sup>2</sup> <sup>≤</sup> *<sup>H</sup>*<sup>2</sup>

The temperature of the liquid in tank 1 is increased by a heater. A temperature sensor continuously monitors the temperature in tank 1, turning the heater on or off. There is also a temperature sensor in tank 2. When *T*<sup>2</sup> decreases, the valve *V*<sup>2</sup> is open and hot liquid comes

*T*1

*T*2

*H*1

*H*2

• Concept *C*1: volume of liquid inside the tank 1 (depends on *V*<sup>1</sup> and *V*2); • Concept *C*2: volume of liquid inside the tank 2 (depends on *V*<sup>1</sup> and *V*2);

Based on this process, a FCM is constructed with eight concepts:

• Concept *C*3: state of *V*<sup>1</sup> (closed, open or partially open); • Concept *C*4: state of *V*<sup>2</sup> (closed, open or partially open); • Concept *C*5: state of *V*<sup>3</sup> (closed, open or partially open);

• Concept *C*6: Temperature of the liquid in tank 1;

Heuristic Search Applied to Fuzzy Cognitive Maps Learning

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225

max; (8a)

max; (8b)

max; (8c)

max. (8d)

$$-0.40 \le W\_{1,3} \le -0.20;\tag{5b}$$

0.20 ≤ *W*1,5 ≤ 0.40; (5c)

$$0.30 \le W\_{2,1} \le 0.40;\tag{5d}$$

$$0.40 \le W\_{3,1} \le 0.50;\tag{5e}$$

$$-1.0 \le W\_{4,1} \le -0.80;\tag{5f}$$

0.50 ≤ *W*5,2 ≤ 0.70; (5g)

$$0.20 \le \mathcal{W}\_{5,4} \le 0.40. \tag{5h}$$

For this problem the following weight matrix is obtained:

$$\mathbf{W} = \begin{bmatrix} 0 & W\_{1,2} & W\_{1,3} & 0 & W\_{1,5} \\ W\_{2,1} & 0 & 0 & 0 & 0 \\ W\_{3,1} & 0 & 0 & 0 & 0 \\ W\_{4,1} & 0 & 0 & 0 & 0 \\ 0 & W\_{5,2} & 0 & W\_{5,4} & 0 \end{bmatrix} . \tag{6}$$

According to [8], all the experts agreed on the range of values for *W*2,1, *W*3,1, and *W*4,1, and most of them agreed on the same range for *W*1,2 and *W*1,3. However, regarding the weights *W*1,5, *W*5,2, and *W*5,4, their opinions varied significantly.

Finally, the group of experts determined that the values output concepts, *C*<sup>1</sup> and *C*5, which are crucial for the system operation, must lie, respectively, in the following regions:

$$0.68 \le A\_1 \le 0.70;\tag{7a}$$

$$0.78 \le A\_5 \le 0.85.\tag{7b}$$

## **2.2. Second control system (PROC2)**

4 Search Algorithms

The experts also had a consensus regarding the range of the weights between concepts, as

0 *W*1,2 *W*1,3 0 *W*1,5 *W*2,1 0000 *W*3,1 0000 *W*4,1 0000 0 *W*5,2 0 *W*5,4 0

According to [8], all the experts agreed on the range of values for *W*2,1, *W*3,1, and *W*4,1, and most of them agreed on the same range for *W*1,2 and *W*1,3. However, regarding the weights

Finally, the group of experts determined that the values output concepts, *C*<sup>1</sup> and *C*5, which

are crucial for the system operation, must lie, respectively, in the following regions:

−0.50 ≤ *W*1,2 ≤ −0.30; (5a) −0.40 ≤ *W*1,3 ≤ −0.20; (5b) 0.20 ≤ *W*1,5 ≤ 0.40; (5c) 0.30 ≤ *W*2,1 ≤ 0.40; (5d) 0.40 ≤ *W*3,1 ≤ 0.50; (5e) −1.0 ≤ *W*4,1 ≤ −0.80; (5f) 0.50 ≤ *W*5,2 ≤ 0.70; (5g) 0.20 ≤ *W*5,4 ≤ 0.40. (5h)

> 

0.68 ≤ *A*<sup>1</sup> ≤ 0.70; (7a) 0.78 ≤ *A*<sup>5</sup> ≤ 0.85. (7b)

. (6)

**Figure 2.** Fuzzy Cognitive Map proposed in [8] for the chemical process control problem.

For this problem the following weight matrix is obtained:

**W** =

*W*1,5, *W*5,2, and *W*5,4, their opinions varied significantly.

 

presented in Equations (5a) to (5h).

In [9] it is considered a system consisting of two identical tanks with one input and one output valve each one, with the output valve of the first tank being the input valve of the second (PROC2), as illustrated in Figure 3.

**Figure 3.** PROC2: a chemical process control problem described in [9].

The objective is to control the volume of liquid within the limits determined by the height *H*min and *H*max and the temperature of the liquid in both tanks within the limits *T*min and *T*max, such that

$$T\_{\min}^1 \le T^1 \le T\_{\max}^1;\tag{8a}$$

$$T\_{\rm min}^2 \le T^2 \le T\_{\rm max}^2;\tag{8b}$$

$$H\_{\min}^1 \le H^1 \le H\_{\max}^1;\tag{8c}$$

$$H\_{\rm min}^2 \le H^2 \le H\_{\rm max}^2. \tag{8d}$$

The temperature of the liquid in tank 1 is increased by a heater. A temperature sensor continuously monitors the temperature in tank 1, turning the heater on or off. There is also a temperature sensor in tank 2. When *T*<sup>2</sup> decreases, the valve *V*<sup>2</sup> is open and hot liquid comes into tank 2.

Based on this process, a FCM is constructed with eight concepts:


According to [9], the fuzzy cognitive map in Figure 4 can be constructed.

**Figure 4.** Fuzzy Cognitive Map proposed in [9] for the chemical process control problem.

It is assumed for PROC2 in this Chapter only causal constraints in the weights between concepts, where concepts *W*4,1 and *W*5,2 are ∈ (−1, 0] and the others have positive causality. The weight matrix for PROC2 is given by

$$\mathbf{W} = \begin{bmatrix} 0 & 0 & W\_{1,3} & W\_{1,4} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & W\_{2,4} & W\_{2,5} & 0 & 0 & 0 \\ W\_{3,1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ W\_{4,1} & W\_{4,2} & 0 & 0 & 0 & 0 & W\_{4,7} & 0 \\ 0 & W\_{5,2} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & W\_{6,3} & 0 & 0 & 0 & W\_{6,8} \\ 0 & 0 & 0 & W\_{7,4} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & W\_{8,6} & 0 & 0 \end{bmatrix} . \tag{9}$$

Finally, the values output concepts, *C*1, *C*2, *C*<sup>6</sup> and *C*7, which are crucial for the system operation, must lie, respectively, in the following regions:

$$0.64 \le A\_1 \le 0.69;\tag{10a}$$

**3. Heuristic FCM learning**

the weights of the FCM.

to the desired output.

3. Hybrid approaches.

• Validation of the adjusted FCM.

classified into three different methods.

1. Hebbian learning based algorithm;

<sup>1</sup> For instance, a valve cannot be negatively open.

A FCM construction can be done in the following manner:

mathematical system model is known.

negative or null) of the causal relationships between concepts.

optimization, differential evolution, simulated annealing, etc;

• Identification of concepts and its interconnections determining the nature (positive,

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• Initial data acquisition by the expert opinions and/or by an equation analysis when the

• Submitting the data from the expert opinions to a fuzzy system which output represents

• Weight adaptation and optimization of the initially proposed FCM, adjusting its response

This section focuses on the weight adaptation (FCM learning). In [16] a very interesting survey on FCM learning is provided. The FCM weights optimization (FCM learning) can be

2. Heuristic optimization techniques, including genetic algorithm, particle swarm

In the Hebbian based methodologies, the FCM weights are iteratively adapted based on a law which depends on the concepts behavior [10], [17]. These algorithms require the experts' knowledge for initial weight values. The differential Hebbian learning (DHL) algorithm proposed by Dickerson and Kosko is a classic example [10]. On the other hand, heuristic (metaheuristic) techniques tries to find a proper **W** matrix by minimizing a cost function based on the error among the desired values of the output concepts and the current output concepts' values (13). The experts' knowledge is not totally necessary, except for the causality constraints, due to the physical restrictions1. These techniques are optimization tools and generally are computationally complex. Examples of hybrid approaching considering

Hebbian learning and Heuristic optimization techniques can be found in [18], [19].

There are several works in the literature dealing with heuristic optimization learning. Most of them are population-based algorithms. For instance, in [8] the PSO algorithm with constriction factor is adopted; in [20] it is presented a FCM learning based on a Tabu Search (TS) and GA combination; in [21] a variation of GA named RCGA (real codec-G.A.) is proposed; in [22] a comparison between GA and Simulated Annealing (SA) is done; in [13] the authors presented a GA based algorithm named Extended Great Deluge Algorithm.

The purpose of the learning is to determine the values of the FCM weights that will produce a desired behavior of the system, which are characterized by *M* output concept values that

$$0.48 \le A\_2 \le 0.52;\tag{10b}$$

$$0.63 \le A\_6 \le 0.67;\tag{10c}$$

$$0.63 \le A\_7 \le 0.67.\tag{10d}$$

Two significant weaknesses of FCMs are its critical dependence on the experts opinions and its potential convergence to undesired states. In order to handle these impairments, learning procedures can be incorporated, increasing the efficiency of FCMs. In this sense, heuristic optimization approach has been deployed as an effective learning method in FCMs [15].

## **3. Heuristic FCM learning**

6 Search Algorithms

• Concept *C*7: Temperature of the liquid in tank 2;

According to [9], the fuzzy cognitive map in Figure 4 can be constructed.

**Figure 4.** Fuzzy Cognitive Map proposed in [9] for the chemical process control problem.

 

It is assumed for PROC2 in this Chapter only causal constraints in the weights between concepts, where concepts *W*4,1 and *W*5,2 are ∈ (−1, 0] and the others have positive causality.

> 0 0 *W*1,3 *W*1,4 0000 000 *W*2,4 *W*2,5 000 *W*3,1 0000000 *W*4,1 *W*4,2 0000 *W*4,7 0 0 *W*5,2 000000 0 0 *W*6,3 0000 *W*6,8 000 *W*7,4 0000 00000 *W*8,6 0 0

Finally, the values output concepts, *C*1, *C*2, *C*<sup>6</sup> and *C*7, which are crucial for the system

Two significant weaknesses of FCMs are its critical dependence on the experts opinions and its potential convergence to undesired states. In order to handle these impairments, learning procedures can be incorporated, increasing the efficiency of FCMs. In this sense, heuristic optimization approach has been deployed as an effective learning method in FCMs [15].

 

0.64 ≤ *A*<sup>1</sup> ≤ 0.69; (10a) 0.48 ≤ *A*<sup>2</sup> ≤ 0.52; (10b) 0.63 ≤ *A*<sup>6</sup> ≤ 0.67; (10c) 0.63 ≤ *A*<sup>7</sup> ≤ 0.67. (10d)

. (9)

• Concept *C*8: Operation of the heater.

The weight matrix for PROC2 is given by

**W** =

 

operation, must lie, respectively, in the following regions:

A FCM construction can be done in the following manner:


This section focuses on the weight adaptation (FCM learning). In [16] a very interesting survey on FCM learning is provided. The FCM weights optimization (FCM learning) can be classified into three different methods.


In the Hebbian based methodologies, the FCM weights are iteratively adapted based on a law which depends on the concepts behavior [10], [17]. These algorithms require the experts' knowledge for initial weight values. The differential Hebbian learning (DHL) algorithm proposed by Dickerson and Kosko is a classic example [10]. On the other hand, heuristic (metaheuristic) techniques tries to find a proper **W** matrix by minimizing a cost function based on the error among the desired values of the output concepts and the current output concepts' values (13). The experts' knowledge is not totally necessary, except for the causality constraints, due to the physical restrictions1. These techniques are optimization tools and generally are computationally complex. Examples of hybrid approaching considering Hebbian learning and Heuristic optimization techniques can be found in [18], [19].

There are several works in the literature dealing with heuristic optimization learning. Most of them are population-based algorithms. For instance, in [8] the PSO algorithm with constriction factor is adopted; in [20] it is presented a FCM learning based on a Tabu Search (TS) and GA combination; in [21] a variation of GA named RCGA (real codec-G.A.) is proposed; in [22] a comparison between GA and Simulated Annealing (SA) is done; in [13] the authors presented a GA based algorithm named Extended Great Deluge Algorithm.

The purpose of the learning is to determine the values of the FCM weights that will produce a desired behavior of the system, which are characterized by *M* output concept values that

<sup>1</sup> For instance, a valve cannot be negatively open.

lie within desired bounds determined by the experts. Hence, the main goal is to obtain a connection (or weight) matrix

$$\mathbf{W} = \begin{bmatrix} \mathbf{W}\_{\mathbf{i}, \mathbf{j}} \end{bmatrix}, \quad \mathbf{i}, \mathbf{j} = \mathbf{1}, \mathbf{2}, \dots, \mathbf{N}, \tag{11}$$

particle is represented as **x**best

from a possible local solution.

**3.2. Genetic Algorithm**

through a cost function.

This Chapter considers the latter case.

(smaller *ω*).

to the following velocity and position equations:

**v***i*(*t* + 1) = *ω* · **v***i*(*t* + 1) + *φ*<sup>1</sup> · **U**1*<sup>i</sup>*

 **x**best

*<sup>g</sup>* (*t*) − **<sup>x</sup>***i*(*t*)

**x***i*(*t* + 1) = **x***i*(*t*) + **v***i*(*t* + 1), (16)

where *φ*<sup>1</sup> and *φ*<sup>2</sup> are two positive constants representing the individual and global acceleration coefficients, respectively, **U**1*<sup>i</sup>* and **U**2*<sup>i</sup>* are diagonal matrices whose elements are random variables uniformly distributed (u.d.) in the interval [0, 1], and *ω* is the inertial weight that plays the role of balancing the global search (higher *ω*) and the local search

A typical value for *φ*<sup>1</sup> and *φ*<sup>2</sup> is *φ*<sup>1</sup> = *φ*<sup>2</sup> = 2 [24]. Regarding the inertia weight, experimental results suggest that it is preferable to initialize *ω* to a large value, and gradually decrease it. The population size P is kept constant in all iterations. In order to obtain further diversification for the search universe, a factor *V*max is added to the PSO model, which is responsible for limiting the velocity in the range [±*V*max], allowing the algorithm to escape

Regarding the FCM, the *i*th candidate vector **x***<sup>i</sup>* is represented by a vector formed by W FCM weights. It is important to point out that after each particle update, restrictions must be imposed on *Wi*,*<sup>j</sup>* according to the experts opinion, before the cost function evaluation.

Genetic Algorithm is an optimization and search technique based on selection mechanism and natural evolution, following Darwin's theory of species' evolution, which explains the history of life through the action of physical processes and genetic operators in populations or species. GA allows a population composed of many individuals to evolve under specified selection rules to a state that maximizes the "fitness" (maximizes or minimizes a cost function). Such an algorithm became popular through the work of John Holland in the early 1970s, and particularly his book Adaptation in Natural and Artificial Systems (1975). The algorithm can be implemented in a binary form or in a continuous (real-valued) form.

Initially, a set of P chromosomes (individuals) is randomly (uniformly distributed) defined, where each chromosome, **x***i*, *i* = 1, 2, ··· , P consists of a vector of variables to be optimized, which, in this case, is formed by FCM weights, respecting the constraints. Each variable is represented by a continuous floating-point number. The P chromosomes are evaluated

*T* strongest chromosomes are selected for mating, generating the mating pool, using the roulette wheel method, where the probability of choosing a given chromosome is proportional to its fitness value. In this work, each pairing generates two offspring with crossover. The weakest *T* chromosomes are changed by the *T* offspring from *T*/2 pairing.

*<sup>i</sup>* . As proposed in [25], the particles are manipulated according

+ *φ*<sup>2</sup> · **U**2*<sup>i</sup>*

 **x**best

Heuristic Search Applied to Fuzzy Cognitive Maps Learning

*<sup>i</sup>* (*t*) <sup>−</sup> **<sup>x</sup>***i*(*t*)

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229

that leads the FCM to a steady state with output concept values within the specified region. Note that, with this notation, and defining **A** = [*A*<sup>1</sup> ··· *AN*] <sup>⊤</sup>, and **<sup>W</sup>** <sup>=</sup> **<sup>W</sup>**<sup>⊤</sup> <sup>+</sup> **<sup>I</sup>**, with {·}<sup>⊤</sup> meaning transposition and **I** identity matrix, Equation (1) can be compactly written as

$$\mathbf{A}(k+1) = f\left(\underline{\mathbf{W}} \cdot \mathbf{A}(k)\right). \tag{12}$$

After the updating procedure in (12), the following cost function is considered for obtaining the optimum weight matrix **W** [8]:

$$F(\mathbf{W}) = \sum\_{i=1}^{M} H\left(\min\left(A\_{\text{out}}^{i}\right) - A\_{\text{out}}^{i}\right) \left|\min\left(A\_{\text{out}}^{i}\right) - A\_{\text{out}}^{i}\right|$$

$$+ \sum\_{i=1}^{M} H\left(A\_{\text{out}}^{i} - \max\left(A\_{\text{out}}^{i}\right)\right) \left|\max\left(A\_{\text{out}}^{i}\right) - A\_{\text{out}}^{i}\right|.\tag{13}$$

where *H* (·) is the Heaviside function, and *A<sup>i</sup>* out, *i* = 1, . . . , *M*, represents the value of the *i*th output concept.

#### **3.1. Particle Swarm Optimization**

The PSO is a meta-heuristic based on the movement of a population (swarm) of individuals (particles) randomly distributed in the search space, each one with its own position and velocity. The position of a particle is modified by the application of velocity in order to reach a better performance [23, 24]. In PSO, each particle is treated as a point in a W-dimensional space2 and represents a candidate vector. The *i*th particle position at instant *t* is represented as

$$\mathbf{x}\_{i}(t) = \begin{bmatrix} \mathbf{x}\_{i,1}(t) \ \mathbf{x}\_{i,2}(t) \ \cdots \ \mathbf{x}\_{i,\mathcal{W}}(t) \end{bmatrix}. \tag{14}$$

In this Chapter, each *xi*,1(*t*) represents one of the *Wi*,*<sup>j</sup>* in the *t*th iteration. Each particle retains a memory of the best position it ever encountered. The best position among all particles until the *t*th iteration (best global position) is represented by **x**best *<sup>g</sup>* , while the best position of the *i*th

<sup>2</sup> W is the number of FCM connections (relationships).

particle is represented as **x**best *<sup>i</sup>* . As proposed in [25], the particles are manipulated according to the following velocity and position equations:

$$\mathbf{v}\_{i}(t+1) = \boldsymbol{\omega} \cdot \mathbf{v}\_{i}(t+1) + \boldsymbol{\phi}\_{1} \cdot \mathbf{U}\_{1i} \left(\mathbf{x}\_{\mathcal{g}}^{\text{best}}(t) - \mathbf{x}\_{i}(t)\right) + \boldsymbol{\phi}\_{2} \cdot \mathbf{U}\_{2i} \left(\mathbf{x}\_{i}^{\text{best}}(t) - \mathbf{x}\_{i}(t)\right) \tag{15}$$

$$\mathbf{x}\_{i}(t+1) = \mathbf{x}\_{i}(t) + \mathbf{v}\_{i}(t+1),\tag{16}$$

where *φ*<sup>1</sup> and *φ*<sup>2</sup> are two positive constants representing the individual and global acceleration coefficients, respectively, **U**1*<sup>i</sup>* and **U**2*<sup>i</sup>* are diagonal matrices whose elements are random variables uniformly distributed (u.d.) in the interval [0, 1], and *ω* is the inertial weight that plays the role of balancing the global search (higher *ω*) and the local search (smaller *ω*).

A typical value for *φ*<sup>1</sup> and *φ*<sup>2</sup> is *φ*<sup>1</sup> = *φ*<sup>2</sup> = 2 [24]. Regarding the inertia weight, experimental results suggest that it is preferable to initialize *ω* to a large value, and gradually decrease it.

The population size P is kept constant in all iterations. In order to obtain further diversification for the search universe, a factor *V*max is added to the PSO model, which is responsible for limiting the velocity in the range [±*V*max], allowing the algorithm to escape from a possible local solution.

Regarding the FCM, the *i*th candidate vector **x***<sup>i</sup>* is represented by a vector formed by W FCM weights. It is important to point out that after each particle update, restrictions must be imposed on *Wi*,*<sup>j</sup>* according to the experts opinion, before the cost function evaluation.

## **3.2. Genetic Algorithm**

8 Search Algorithms

connection (or weight) matrix

the optimum weight matrix **W** [8]:

*F*(**W**) =

*M* ∑ *i*=1 *H* min *Ai* out − *A<sup>i</sup>* out min *Ai* out − *A<sup>i</sup>* out 

+ *M* ∑ *i*=1 *H Ai*

**<sup>x</sup>***i*(*t*) =

the *t*th iteration (best global position) is represented by **x**best

<sup>2</sup> W is the number of FCM connections (relationships).

where *H* (·) is the Heaviside function, and *A<sup>i</sup>*

**3.1. Particle Swarm Optimization**

output concept.

as

lie within desired bounds determined by the experts. Hence, the main goal is to obtain a

that leads the FCM to a steady state with output concept values within the specified region.

After the updating procedure in (12), the following cost function is considered for obtaining

 *Ai* out max *Ai* out − *A<sup>i</sup>* out 

The PSO is a meta-heuristic based on the movement of a population (swarm) of individuals (particles) randomly distributed in the search space, each one with its own position and velocity. The position of a particle is modified by the application of velocity in order to reach a better performance [23, 24]. In PSO, each particle is treated as a point in a W-dimensional space2 and represents a candidate vector. The *i*th particle position at instant *t* is represented

*xi*,1(*t*) *xi*,2(*t*) ··· *xi*,<sup>W</sup> (*t*)

In this Chapter, each *xi*,1(*t*) represents one of the *Wi*,*<sup>j</sup>* in the *t*th iteration. Each particle retains a memory of the best position it ever encountered. The best position among all particles until

out − max

meaning transposition and **I** identity matrix, Equation (1) can be compactly written as

, *i*, *j* = 1, 2, . . . , *N*, (11)

**A**(*k* + 1) = *f* (**W** · **A**(*k*)). (12)

out, *i* = 1, . . . , *M*, represents the value of the *i*th

<sup>⊤</sup>, and **<sup>W</sup>** <sup>=</sup> **<sup>W</sup>**<sup>⊤</sup> <sup>+</sup> **<sup>I</sup>**, with {·}<sup>⊤</sup>

, (13)

. (14)

*<sup>g</sup>* , while the best position of the *i*th

**W** = *Wi*,*<sup>j</sup>* 

Note that, with this notation, and defining **A** = [*A*<sup>1</sup> ··· *AN*]

Genetic Algorithm is an optimization and search technique based on selection mechanism and natural evolution, following Darwin's theory of species' evolution, which explains the history of life through the action of physical processes and genetic operators in populations or species. GA allows a population composed of many individuals to evolve under specified selection rules to a state that maximizes the "fitness" (maximizes or minimizes a cost function). Such an algorithm became popular through the work of John Holland in the early 1970s, and particularly his book Adaptation in Natural and Artificial Systems (1975). The algorithm can be implemented in a binary form or in a continuous (real-valued) form. This Chapter considers the latter case.

Initially, a set of P chromosomes (individuals) is randomly (uniformly distributed) defined, where each chromosome, **x***i*, *i* = 1, 2, ··· , P consists of a vector of variables to be optimized, which, in this case, is formed by FCM weights, respecting the constraints. Each variable is represented by a continuous floating-point number. The P chromosomes are evaluated through a cost function.

*T* strongest chromosomes are selected for mating, generating the mating pool, using the roulette wheel method, where the probability of choosing a given chromosome is proportional to its fitness value. In this work, each pairing generates two offspring with crossover. The weakest *T* chromosomes are changed by the *T* offspring from *T*/2 pairing. The crossover procedure is similar to the one presented in [26]. It begins by randomly selecting a variable in the first pair of parents to be the crossover point

$$\mathfrak{a} = \lceil \mathfrak{u} \cdot \mathcal{W} \rceil \; , \tag{17}$$

In the classical DE, a perturbation is created by using a difference vector based mutation,

where the real and constant factor *Fe* (typically ∈ [0.5, 1.0]) controls the gain of the differential variation. The indexes *r*0, *r*1 and *r*2 are randomly chosen and mutually exclusive. In this work, an alternative perturbation procedure named DE/current-to-best/1/bin is

There are also other variants of the perturbation procedure [28, 29]. A uniform crossover operation is applied in order to have diversity enhancement, which mixes parameters of the

where *χ* is the crossover constant typically ∈ [0.8, 1.0] and *r* is a random variable u.d. in the interval [0, 1). In order to prevent the case **u***<sup>i</sup>* = **x***i*, at least one component is taken from the

For selecting, the algorithm uses a simple method where the trial vector **u***<sup>i</sup>* competes against

**<sup>x</sup>***i*(*<sup>t</sup>* <sup>+</sup> <sup>1</sup>) = **<sup>u</sup>***i*(*t*) if *<sup>f</sup>* (**u***i*(*t*)) <sup>≤</sup> *<sup>f</sup>* (**x***i*(*t*))

All the simulations were taken considering 10<sup>4</sup> trials for PROC1 and PROC2. It is worth noting that PSO, GA and DE input parameters were chosen previously after exhaustive simulation tests. As a result, optimal or quasi-optimal input parameters for the PSO, GA

The adopted values for the input parameters of PSO, GA and DE are summarized on Table 1. DE has less parameter inputs than the other two methods, which is a relative advantage. Four different scenarios for the process PROC1 were analyzed. The main performance results

This scenario considers all the constraints on the FCM weights shown in Equations (5a) to

(5h). As mentioned in [8] and also verified here, there is no solution in this case.

mutation vector **y***<sup>i</sup>* and **x***i*, for generating the trial vector **u***i*:

**<sup>y</sup>***<sup>i</sup>* if(*<sup>r</sup>* ≤ *<sup>χ</sup>*)

**u***<sup>i</sup>* =

and DE heuristic algorithms have been obtained.

for each scenario is described in the next subsections.

considered [28, 29], such that

mutation vector **y***i*.

the target vector **x***i*, such that,

**4. Simulation results**

**4.1. Process PROC1**

*4.1.1. Scenario 1*

**y***<sup>i</sup>* = **x***r*<sup>0</sup> + *Fe*·(**x***r*<sup>1</sup> − **x***r*2), *i* = 1, 2, ··· , P, (22)

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**<sup>x</sup>***<sup>i</sup>* otherwise *<sup>i</sup>* <sup>=</sup> 1, 2, ··· , <sup>P</sup>, (24)

**<sup>x</sup>***i*(*t*) otherwise (25)

**<sup>y</sup>***<sup>i</sup>* = **<sup>x</sup>***<sup>i</sup>* + *Fe* · (**x**best − **<sup>x</sup>***i*) + *Fe* · (**x***r*<sup>1</sup> − **<sup>x</sup>***r*2), *<sup>i</sup>* = 1, 2, ··· , P, (23)

where *u* is an u.d. random variable (r.v.) in the interval [0, 1], and ⌈·⌉ is the upper integer operator. The *j*th pair of parents, *j* = 1, 2, ··· , *T*/2 is defined as

$$\begin{array}{l} \mathbf{d} \mathbf{ad}\_{\dot{j}} = \begin{bmatrix} \boldsymbol{\mathfrak{x}}\_{j,1}^{d} \ \boldsymbol{\mathfrak{x}}\_{j,2}^{d} \ \cdots \ \boldsymbol{\mathfrak{x}}\_{j,\boldsymbol{\mathfrak{x}}}^{d} \ \cdots \ \boldsymbol{\mathfrak{x}}\_{j,\boldsymbol{\mathcal{W}}}^{d} \end{bmatrix} \\\mathbf{m} \mathbf{om}\_{\dot{j}} = \begin{bmatrix} \boldsymbol{\mathfrak{x}}\_{j,1}^{m} \ \boldsymbol{\mathfrak{x}}\_{j,2}^{m} \ \cdots \ \boldsymbol{\mathfrak{x}}\_{j,\boldsymbol{\mathfrak{x}}}^{m} \ \cdots \ \boldsymbol{\mathfrak{x}}\_{j,\boldsymbol{\mathcal{W}}}^{m} \end{bmatrix} . \end{array} \tag{18}$$

Then the selected variables are combined to form new variables that will appear in the offspring

$$\begin{aligned} \mathbf{x}^{\alpha}\_{j,1} &= \mathbf{x}^{m}\_{j,\alpha} - \beta \left[ \mathbf{x}^{m}\_{j,\alpha} - \mathbf{x}^{d}\_{j,\alpha} \right]; \\ \mathbf{x}^{\alpha}\_{j,2} &= \mathbf{x}^{d}\_{j,\alpha} + \beta \left[ \mathbf{x}^{m}\_{j,\alpha} - \mathbf{x}^{d}\_{j,\alpha} \right]. \end{aligned} \tag{19}$$

where *β* is also a r.v. u.d. in the interval [0, 1]. Finally,

$$\begin{array}{c} \text{offsfspring}\_{1} = \begin{bmatrix} \mathbf{x}\_{j,1}^{d} \ \mathbf{x}\_{j,2}^{d} \ \mathbf{x}\_{j,a-1}^{d} \ \cdots \ \mathbf{x}\_{j,1}^{o} \ \mathbf{x}\_{j,a+1}^{m} \ \cdots \ \mathbf{x}\_{j,\mathcal{W}}^{m} \end{bmatrix} \\ \text{offsfspring}\_{2} = \begin{bmatrix} \mathbf{x}\_{j,1}^{m} \ \mathbf{x}\_{j,2}^{m} \ \mathbf{x}\_{j,a-1}^{m} \ \cdots \ \mathbf{x}\_{j,2}^{o} \ \mathbf{x}\_{j,a+1}^{d} \ \cdots \ \mathbf{x}\_{j,\mathcal{W}}^{d} \end{bmatrix} . \end{array} \tag{20}$$

In order to allow escaping from possible local minima, a mutation operation is introduced in the resultant population, except for the strongest one (elitism). It is assumed in this work a Gaussian mutation. If the rate of mutations is given by *Pm*, there will be *Nm* = ⌈*Pm* · (P − 1) ·W⌉ mutations uniformly chosen among (P − 1) · W variables. If *xi*,*<sup>w</sup>* is chosen, with *w* = 1, 2, ··· , W, than, after Gaussian mutation, it is substituted by

$$\mathbf{x}'\_{i,w} = \mathbf{x}\_{i,w} + \mathcal{N}\left(\mathbf{0}, \sigma\_m^2\right),\tag{21}$$

where N 0, *σ*<sup>2</sup> *m* represents a normal r.v. with zero mean and variance *σ*<sup>2</sup> *m*.

After mutation, restrictions must be imposed on *Wi*,*<sup>j</sup>* according to the experts opinion, before the cost function evaluation.

#### **3.3. Differential Evolution**

The Differential Evolution (DE) search has been introduced by Ken Price and Rainer Storn [27, 28]. DE is a parallel direct search method. As in GA, a population with P elements is randomly defined, where each W-dimension element consists of a **x** vector of variables to be optimized (FCM weights in this case) respecting the constraints.

In the classical DE, a perturbation is created by using a difference vector based mutation,

$$\mathbf{y}\_{i} = \mathbf{x}\_{r0} + F\_{\ell} \cdot (\mathbf{x}\_{r1} - \mathbf{x}\_{r2}) \,, \qquad \qquad \mathbf{i} = 1, 2, \cdots, r, \mathcal{P}\_{r} \tag{22}$$

where the real and constant factor *Fe* (typically ∈ [0.5, 1.0]) controls the gain of the differential variation. The indexes *r*0, *r*1 and *r*2 are randomly chosen and mutually exclusive. In this work, an alternative perturbation procedure named DE/current-to-best/1/bin is considered [28, 29], such that

$$\mathbf{y}\_{i} = \mathbf{x}\_{i} + F\_{\varepsilon} \cdot \left(\mathbf{x}\_{\text{best}} - \mathbf{x}\_{i}\right) + F\_{\varepsilon} \cdot \left(\mathbf{x}\_{r1} - \mathbf{x}\_{r2}\right), \qquad \qquad \text{i} = 1, 2, \dots, \mathcal{P}, \tag{23}$$

There are also other variants of the perturbation procedure [28, 29]. A uniform crossover operation is applied in order to have diversity enhancement, which mixes parameters of the mutation vector **y***<sup>i</sup>* and **x***i*, for generating the trial vector **u***i*:

$$\mathbf{u}\_{i} = \begin{cases} \mathbf{y}\_{i} & \text{if } (r \le \chi) \\ \mathbf{x}\_{i} \text{ otherwise} & \end{cases} \qquad \qquad i = 1, 2, \cdots, \mathcal{P}\_{\prime} \tag{24}$$

where *χ* is the crossover constant typically ∈ [0.8, 1.0] and *r* is a random variable u.d. in the interval [0, 1). In order to prevent the case **u***<sup>i</sup>* = **x***i*, at least one component is taken from the mutation vector **y***i*.

For selecting, the algorithm uses a simple method where the trial vector **u***<sup>i</sup>* competes against the target vector **x***i*, such that,

$$\mathbf{x}\_{i}(t+1) = \begin{cases} \mathbf{u}\_{i}(t) \text{ if } f\left(\mathbf{u}\_{i}(t)\right) \le f\left(\mathbf{x}\_{i}(t)\right) \\ \mathbf{x}\_{i}(t) \qquad \text{otherwise} \end{cases} \tag{25}$$

## **4. Simulation results**

10 Search Algorithms

offspring

where N

0, *σ*<sup>2</sup> *m* 

the cost function evaluation.

**3.3. Differential Evolution**

The crossover procedure is similar to the one presented in [26]. It begins by randomly

where *u* is an u.d. random variable (r.v.) in the interval [0, 1], and ⌈·⌉ is the upper integer

*<sup>j</sup>*,2 ··· *<sup>x</sup><sup>d</sup>*

Then the selected variables are combined to form new variables that will appear in the

*<sup>j</sup>*,*<sup>α</sup>* <sup>−</sup> *<sup>β</sup> xm <sup>j</sup>*,*<sup>α</sup>* <sup>−</sup> *<sup>x</sup><sup>d</sup> j*,*α* ;

*<sup>j</sup>*,*<sup>α</sup>* + *β xm <sup>j</sup>*,*<sup>α</sup>* <sup>−</sup> *<sup>x</sup><sup>d</sup> j*,*α* 

*<sup>j</sup>*,2 ··· *<sup>x</sup><sup>m</sup>*

*<sup>j</sup>*,*α*−<sup>1</sup> ··· *<sup>x</sup>*<sup>o</sup>

*<sup>j</sup>*,*α*−<sup>1</sup> ··· *<sup>x</sup>*<sup>o</sup>

 0, *σ*<sup>2</sup> *m* 

In order to allow escaping from possible local minima, a mutation operation is introduced in the resultant population, except for the strongest one (elitism). It is assumed in this work a Gaussian mutation. If the rate of mutations is given by *Pm*, there will be *Nm* = ⌈*Pm* · (P − 1) ·W⌉ mutations uniformly chosen among (P − 1) · W variables. If *xi*,*<sup>w</sup>* is

chosen, with *w* = 1, 2, ··· , W, than, after Gaussian mutation, it is substituted by

*<sup>i</sup>*,*<sup>w</sup>* <sup>=</sup> *xi*,*<sup>w</sup>* <sup>+</sup> <sup>N</sup>

represents a normal r.v. with zero mean and variance *σ*<sup>2</sup>

After mutation, restrictions must be imposed on *Wi*,*<sup>j</sup>* according to the experts opinion, before

The Differential Evolution (DE) search has been introduced by Ken Price and Rainer Storn [27, 28]. DE is a parallel direct search method. As in GA, a population with P elements is randomly defined, where each W-dimension element consists of a **x** vector of variables to

*<sup>j</sup>*,1 *<sup>x</sup><sup>m</sup>*

*<sup>j</sup>*,2 *<sup>x</sup><sup>d</sup>*

*<sup>j</sup>*,*α*+<sup>1</sup> ··· *<sup>x</sup><sup>m</sup>*

*<sup>j</sup>*,*α*+<sup>1</sup> ··· *<sup>x</sup><sup>d</sup>*

*j*,W 

*j*,W 

*<sup>j</sup>*,*<sup>α</sup>* ··· *<sup>x</sup><sup>d</sup>*

*<sup>j</sup>*,*<sup>α</sup>* ··· *<sup>x</sup><sup>m</sup> j*,W

*j*,W 

*α* = ⌈*u* · W⌉ , (17)

. (18)

. (19)

, (21)

*m*.

. (20)

selecting a variable in the first pair of parents to be the crossover point

operator. The *j*th pair of parents, *j* = 1, 2, ··· , *T*/2 is defined as

dad*<sup>j</sup>* =

mom*<sup>j</sup>* =

*x*o *<sup>j</sup>*,1 <sup>=</sup> *<sup>x</sup><sup>m</sup>*

*x*o *<sup>j</sup>*,2 <sup>=</sup> *<sup>x</sup><sup>d</sup>*

 *xd <sup>j</sup>*,1 *<sup>x</sup><sup>d</sup> <sup>j</sup>*,2 *<sup>x</sup><sup>d</sup>*

 *xm <sup>j</sup>*,1 *<sup>x</sup><sup>m</sup> <sup>j</sup>*,2 *<sup>x</sup><sup>m</sup>*

*x*′

be optimized (FCM weights in this case) respecting the constraints.

where *β* is also a r.v. u.d. in the interval [0, 1]. Finally,

offspring1 =

offspring2 =

 *xd <sup>j</sup>*,1 *<sup>x</sup><sup>d</sup>*

 *xm <sup>j</sup>*,1 *<sup>x</sup><sup>m</sup>*

> All the simulations were taken considering 10<sup>4</sup> trials for PROC1 and PROC2. It is worth noting that PSO, GA and DE input parameters were chosen previously after exhaustive simulation tests. As a result, optimal or quasi-optimal input parameters for the PSO, GA and DE heuristic algorithms have been obtained.

#### **4.1. Process PROC1**

The adopted values for the input parameters of PSO, GA and DE are summarized on Table 1. DE has less parameter inputs than the other two methods, which is a relative advantage. Four different scenarios for the process PROC1 were analyzed. The main performance results for each scenario is described in the next subsections.

#### *4.1.1. Scenario 1*

This scenario considers all the constraints on the FCM weights shown in Equations (5a) to (5h). As mentioned in [8] and also verified here, there is no solution in this case.


**GA with** P = 10 *A*max 0.6882 0.8051 0.6183 0.6964 0.8186 *A*min 0.6800 0.7297 0.5749 0.6590 0.7800 *A*average 0.6833 0.7726 0.6123 0.6605 0.8059

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**DE with** P = 10 *A*max 0.6882 0.8062 0.6217 0.8389 0.8186 *A*min 0.6243 0.5563 0.5749 0.6590 0.6590 *A*average 0.6746 0.7685 0.6074 0.6620 0.8078

**DE with** P = 20 *A*max 0.6882 0.8058 0.6203 0.8389 0.8186 *A*min 0.6477 0.5563 0.5749 0.6590 0.7776 *A*average 0.6810 0.7845 0.6070 0.6616 0.8089

**DE with** P = 30 *A*max 0.6882 0.8054 0.6192 0.6965 0.8186 *A*min 0.6653 0.5984 0.5749 0.6590 0.7800 *A*average 0.6819 0.7861 0.6072 0.6614 0.8089

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> 0.5

(b)

Number of Generations

C1 C2 C3 C4 C5 0.55 0.6 0.65 0.7 0.75 0.8 0.85

Concept Values

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> 0.5

(c)

Number of Iterations

C1 C2 C3 C4 C5

Number of Failures: 0 Probability of Success: 1.0

Number of Failures: 4372 Probability of Success: 0.5628

Number of Failures: 537 Probability of Success: 0.9463

Number of Failures: 63 Probability of Success: 0.9937

> C1 C2 C3 C4 C5

0.55 0.6 0.65 0.7 0.75 0.8 0.85

**Figure 5.** Mean convergence for (a) PSO, (b) GA and (c) DE in PROC1, Scenario 2 with P = 10.

Concept Values

**Table 4.** DE simulation results for Scenario 2, PROC1.

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> 0.5

(a)

Number of Iterations

0.55 0.6 0.65 0.7 0.75 0.8 0.85

Concept Values

**Table 3.** GA simulation results for Scenario 2, PROC1.

**Table 1.** PSO, GA and DE input parameters values for PROC1.

#### *4.1.2. Scenario 2*

In this scenario, the constraints on the FCM weights *W*1,5, *W*5,2, and *W*5,4 have been relaxed, since the experts' opinions have varied significantly. In this case the values of these weights were allowed to assume any value in the interval [0, 1] in order to keep the causality of relationships. Tables 2, 3 and 4 present the obtained simulation results for PSO, GA and DE, respectively.

As can be observed, in the Scenario 2, GA has a better performance than DE and PSO, achieving convergence without failure with P = 10. PSO has the second best performance, presenting no convergence errors in 10<sup>4</sup> independent experiments when P = 20. DE does not achieve 100% of performance success, even for P = 30, when 63 errors occurred in 104 independent experiments. Figures 5 and 6 present the mean FCM concepts convergence in the Scenario 2 under 104 trials. Considering the best case simulated for each algorithm (GA with P = 10, PSO with P = 20 and DE with P = 30), DE and PSO presented similar average convergence, being faster than GA.


**Table 2.** PSO simulation results for Scenario 2, PROC1.


**Table 3.** GA simulation results for Scenario 2, PROC1.

12 Search Algorithms

*4.1.2. Scenario 2*

respectively.

**PSO** Population P = 10, 20, 30 Acceleration Coefficients *φ*<sup>1</sup> = 2, *φ*<sup>2</sup> = 0.2

**GA**

Mating Pool *T* = 2 · round(P/4)

**DE** Population P = 10, 20, 30

In this scenario, the constraints on the FCM weights *W*1,5, *W*5,2, and *W*5,4 have been relaxed, since the experts' opinions have varied significantly. In this case the values of these weights were allowed to assume any value in the interval [0, 1] in order to keep the causality of relationships. Tables 2, 3 and 4 present the obtained simulation results for PSO, GA and DE,

As can be observed, in the Scenario 2, GA has a better performance than DE and PSO, achieving convergence without failure with P = 10. PSO has the second best performance, presenting no convergence errors in 10<sup>4</sup> independent experiments when P = 20. DE does not achieve 100% of performance success, even for P = 30, when 63 errors occurred in 104 independent experiments. Figures 5 and 6 present the mean FCM concepts convergence in the Scenario 2 under 10<sup>4</sup> trials. Considering the best case simulated for each algorithm (GA with P = 10, PSO with P = 20 and DE with P = 30), DE and PSO presented similar average

> **PSO with** P = 10 *A*max 0.6882 0.8058 0.6203 0.8389 0.8186 *A*min 0.6477 0.7297 0.5749 0.6590 0.7800 *A*average 0.6840 0.7911 0.6178 0.6713 0.8148

> **PSO with** P = 20 *A*max 0.6882 0.8051 0.6183 0.6967 0.8186 *A*min 0.6800 0.7297 0.5750 0.6590 0.7801 *A*average 0.6838 0.7926 0.6178 0.6730 0.8139

Inertial Weight *ω* = 1.2 Initial Velocity **v***i*(0) = 0.1 Maximum Velocity *V*max = 2

Population P = 10

Rate of Mutation *Pm* = 0.2 Mutation Std. Deviation *σ<sup>m</sup>* = 0.3

Crossover Constant *χ*= 0.9 DE Gain Variation Factor *Fe* = 0.9

**Table 1.** PSO, GA and DE input parameters values for PROC1.

convergence, being faster than GA.

**Table 2.** PSO simulation results for Scenario 2, PROC1.

Number of Failures: 49 Probability of Success: 0.9951

Number of Failures: 0 Probability of Success: 1.0


**Table 4.** DE simulation results for Scenario 2, PROC1.

**Figure 5.** Mean convergence for (a) PSO, (b) GA and (c) DE in PROC1, Scenario 2 with P = 10.

**PSO with** P = 10 *A*max 0.7000 0.8404 0.6590 0.8404 0.8206 *A*min 0.6800 0.4339 0.4339 0.6590 0.7800 *A*average 0.6907 0.6886 0.6112 0.7333 0.8080

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**GA with** P = 10 *A*max 0.7000 0.8404 0.6590 0.8404 0.8206 *A*min 0.6800 0.4339 0.4339 0.6590 0.7800 *A*average 0.6906 0.6496 0.5914 0.7139 0.8027

**DE with** P = 10 *A*max 0.7189 0.8404 0.6590 0.8404 0.8206 *A*min 0.6590 0.4294 0.4277 0.6590 0.6590 *A*average 0.6896 0.6560 0.5839 0.7209 0.8060

**DE with** P = 20 *A*max 0.7000 0.8404 0.6590 0.8404 0.8206 *A*min 0.6800 0.4339 0.4339 0.6590 0.7800 *A*average 0.6897 0.6557 0.5848 0.7204 0.8074

In this situation, the causality and the strength of the causality were totally relaxed. PSO and GA algorithms were able to determine proper weights for the connection matrix with P = 10, and DE did not have success in only 4 experiments, resulting in a probability of success equal to 0.9996. When P = 20, DE did not presented convergence errors. Table 6 presents the obtained results, while Figure 8 presents the mean convergence of the concepts in 10<sup>4</sup> independent experiments. As in Scenario 3, DE presented the fastest average convergence,

For this process the value of the weights were fixed in the interval [0, 1] or in the interval [−1, 0), according to the causality determined by the experts. The adopted input simulation parameters for PSO, GA and DE algorithms were the same considered in PROC1, except for

Table 7 summarizes the simulation results, while Figures 9 and 10 present the mean convergence of the concepts value considering 104 independent experiments. As can be seen, P = 10 was enough for achieving convergence rate of 100% in GA. With P = 10, PSO presented 30 failures and probability of success equal to 0.9970, while DE presented 897 failures and probability of success equal to 0.9103. However, with P = 20, both PSO and DE were able to achieve 100% of success. For this process, DE has a faster average convergence

Number of Failures: 0 Probability of Success: 1.0

Number of Failures: 0 Probability of Success: 1.0

Number of Failures: 36 Probability of Success: 0.9964

Number of Failures: 0 Probability of Success: 1.0

while GA and PSO presented similar average convergence speed.

the population size in PSO that was assumed P = 10 and P = 20.

**Table 5.** GA, PSO and DE simulation results for Scenario 3, PROC1.

than PSO and GA, being PSO the second fastest.

*4.1.4. Scenario 4*

**4.2. Process PROC2**

**Figure 6.** Mean convergence in PROC1, Scenario 2 for (a) PSO with P = 20, (b) DE with P = 20 and (c) DE with P = 30.

#### *4.1.3. Scenario 3*

In this Scenario, all the weights constrains were relaxed, but the causalities were kept, i.e., the value of the weights were fixed in the interval [0, 1] or in the interval [−1, 0), according to the causality determined by the experts. Table 5 presents the obtained results. As can be seen, P = 10 was enough for achieving 100% of convergence in GA and PSO. With P = 10, DE was not able to find the proper solution in 36 trials, resulting in a probability of success equal to 0.9964, but when P = 20, DE obtained 100% of success. Figure 7 presents the mean convergence of the concepts in 104 independent experiments. In this scenario, DE presented the fastest average convergence.

**Figure 7.** Mean convergence for Scenario 3 in PROC1 considering (a) PSO with P = 10, (b) GA with P = 10, (c) DE with P = 10, and (d) DE with P = 20.


**Table 5.** GA, PSO and DE simulation results for Scenario 3, PROC1.

#### *4.1.4. Scenario 4*

14 Search Algorithms

0.55 0.6 0.65 0.7 0.75 0.8 0.85

0.55 0.6 0.65 0.7 0.75 0.8 0.85

Concept Values

*4.1.3. Scenario 3*

Concept Values

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> 0.5

(a)

Number of Iterations

the fastest average convergence.

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> 0.5

(a)

P = 10, and (d) DE with P = 20.

Number of Iterations

C1 C2 C3 C4 C5

> C1 C2 C3 C4 C5

0.55 0.6 0.65 0.7 0.75 0.8 0.85

0.55 0.6 0.65 0.7 0.75 0.8 0.85

Concept Values

Concept Values

0.55 0.6 0.65 0.7 0.75 0.8 0.85

Concept Values

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> 0.5

(b)

In this Scenario, all the weights constrains were relaxed, but the causalities were kept, i.e., the value of the weights were fixed in the interval [0, 1] or in the interval [−1, 0), according to the causality determined by the experts. Table 5 presents the obtained results. As can be seen, P = 10 was enough for achieving 100% of convergence in GA and PSO. With P = 10, DE was not able to find the proper solution in 36 trials, resulting in a probability of success equal to 0.9964, but when P = 20, DE obtained 100% of success. Figure 7 presents the mean convergence of the concepts in 104 independent experiments. In this scenario, DE presented

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> 0.5

(b)

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> 0.5

(d)

**Figure 7.** Mean convergence for Scenario 3 in PROC1 considering (a) PSO with P = 10, (b) GA with P = 10, (c) DE with

Number of Iterations

Number of Generations

**Figure 6.** Mean convergence in PROC1, Scenario 2 for (a) PSO with P = 20, (b) DE with P = 20 and (c) DE with P = 30.

Number of Iterations

C1 C2 C3 C4 C5

> C1 C2 C3 C4 C5

C1 C2 C3 C4 C5 0.55 0.6 0.65 0.7 0.75 0.8 0.85

Concept Values

0.55 0.6 0.65 0.7 0.75 0.8 0.85

Concept Values

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> 0.5

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> 0.5

(c)

Number of Iterations

(c)

Number of Iterations

C1 C2 C3 C4 C5

C1 C2 C3 C4 C5

> In this situation, the causality and the strength of the causality were totally relaxed. PSO and GA algorithms were able to determine proper weights for the connection matrix with P = 10, and DE did not have success in only 4 experiments, resulting in a probability of success equal to 0.9996. When P = 20, DE did not presented convergence errors. Table 6 presents the obtained results, while Figure 8 presents the mean convergence of the concepts in 10<sup>4</sup> independent experiments. As in Scenario 3, DE presented the fastest average convergence, while GA and PSO presented similar average convergence speed.

#### **4.2. Process PROC2**

For this process the value of the weights were fixed in the interval [0, 1] or in the interval [−1, 0), according to the causality determined by the experts. The adopted input simulation parameters for PSO, GA and DE algorithms were the same considered in PROC1, except for the population size in PSO that was assumed P = 10 and P = 20.

Table 7 summarizes the simulation results, while Figures 9 and 10 present the mean convergence of the concepts value considering 104 independent experiments. As can be seen, P = 10 was enough for achieving convergence rate of 100% in GA. With P = 10, PSO presented 30 failures and probability of success equal to 0.9970, while DE presented 897 failures and probability of success equal to 0.9103. However, with P = 20, both PSO and DE were able to achieve 100% of success. For this process, DE has a faster average convergence than PSO and GA, being PSO the second fastest.

**PSO with** P = 10 *A*max 0.7000 0.9199 0.8206 0.8404 0.8206 *A*min 0.6800 0.2129 0.4339 0.3952 0.7800 *A*average 0.6901 0.6908 0.6767 0.6777 0.8101

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**GA with** P = 10 *A*max 0.7000 0.9192 0.8206 0.8404 0.8206 *A*min 0.6800 0.2130 0.4339 0.3952 0.7800 *A*average 0.6905 0.6154 0.6371 0.6306 0.8030

**DE with** P = 10 *A*max 0.7166 0.9199 0.8206 0.8404 0.8229 *A*min 0.6800 0.2112 0.4285 0.3948 0.7628 *A*average 0.6901 0.6087 0.6358 0.6232 0.8084

**DE with** P = 20 *A*max 0.7000 0.9199 0.8206 0.8404 0.8206 *A*min 0.6800 0.2129 0.4338 0.3952 0.7800 *A*average 0.6902 0.6061 0.6311 0.6227 0.8102

There are in the literature, several models based on fuzzy cognitive maps developed and adapted to a large range of applications. The use of learning algorithms may be necessary for obtaining proper values for the weight matrix and reducing the dependence on the experts'

Within this context, this Chapter presented a comparison of three heuristic search approaches, PSO, GA and DE, applied to FCM weight optimization in two processes control. In all the considered cases, GA presented a performance in terms of probability of success better or equal to the other two schemes, being PSO the second best technique in terms of

Specially in scenario 2 of PROC 1, when there are several weight constraints, GA achieved 100% of success in 10<sup>4</sup> independent experiments with a population of 10 chromosomes. PSO needed P = 20 particles in the population in order to reach 100% of success. DE was not

Number of Failures: 0 Probability of Success: 1.0

Number of Failures: 0 Probability of Success: 1.0

Number of Failures: 4

Number of Failures: 0 Probability of Success: 1.0

**Table 6.** GA, PSO and DE simulation results for Scenario 4, PROC1.

probability of success in the two considered processes.

able to achieve 100% of success even for P = 30.

**5. Conclusions**

knowledge.

Probability of Success: 0.9996

**Figure 8.** Mean convergence for Scenario 4 in PROC1 considering (a) PSO with P = 10, (b) GA with P = 10, (c) DE with P = 10, and (d) DE with P = 20.

**Figure 9.** Mean convergence for (a) PSO, (b) GA and (c) DE in PROC2 with P = 10.

**Figure 10.** Mean convergence for (a) PSO and (b) DE in PROC2 with P = 20.


**Table 6.** GA, PSO and DE simulation results for Scenario 4, PROC1.

## **5. Conclusions**

16 Search Algorithms

0.55 0.6 0.65 0.7 0.75 0.8 0.85

0.5 0.55 0.6 0.65 0.7 0.75 0.8

Concept Values

Concept Values

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> 0.5

(a)

P = 10, and (d) DE with P = 20.

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> 0.45

(a)

0.5 0.55 0.6 0.65 0.7 0.75 0.8

Concept Values

Number of Iterations

Number of Iterations

C1 C2 C3 C4 C5 0.55 0.6 0.65 0.7 0.75 0.8 0.85

0.55 0.6 0.65 0.7 0.75 0.8 0.85

> 0.5 0.55 0.6 0.65 0.7 0.75 0.8

Concept Values

**Figure 9.** Mean convergence for (a) PSO, (b) GA and (c) DE in PROC2 with P = 10.

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> 0.45

(a)

Number of Iterations

**Figure 10.** Mean convergence for (a) PSO and (b) DE in PROC2 with P = 20.

Concept Values

C1 C2 C3 C4 C5 C6 C7 C8 Concept Values

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> 0.5

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> 0.5

(d)

**Figure 8.** Mean convergence for Scenario 4 in PROC1 considering (a) PSO with P = 10, (b) GA with P = 10, (c) DE with

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> 0.45

(b)

C1 C2 C3 C4 C5 C6 C7 C8

Number of Generations

0.5 0.55 0.6 0.65 0.7 0.75 0.8

Concept Values

Number of Iterations

(b)

Number of Generations

C1 C2 C3 C4 C5

C1 C2 C3 C4 C5

C1 C2 C3 C4 C5 C6 C7 C8

0.5 0.55 0.6 0.65 0.7 0.75 0.8

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> 0.45

(b)

Number of Iterations

Concept Values

0.55 0.6 0.65 0.7 0.75 0.8 0.85

Concept Values

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> 0.5

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> 0.45

(c)

Number of Iterations

C1 C2 C3 C4 C5 C6 C7 C8

(c)

Number of Iterations

C1 C2 C3 C4 C5

> C1 C2 C3 C4 C5 C6 C7 C8

> > There are in the literature, several models based on fuzzy cognitive maps developed and adapted to a large range of applications. The use of learning algorithms may be necessary for obtaining proper values for the weight matrix and reducing the dependence on the experts' knowledge.

> > Within this context, this Chapter presented a comparison of three heuristic search approaches, PSO, GA and DE, applied to FCM weight optimization in two processes control. In all the considered cases, GA presented a performance in terms of probability of success better or equal to the other two schemes, being PSO the second best technique in terms of probability of success in the two considered processes.

> > Specially in scenario 2 of PROC 1, when there are several weight constraints, GA achieved 100% of success in 10<sup>4</sup> independent experiments with a population of 10 chromosomes. PSO needed P = 20 particles in the population in order to reach 100% of success. DE was not able to achieve 100% of success even for P = 30.


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**Table 7.** GA, PSO and DE simulation results for PROC2.

In most scenarios, DE has presented a faster convergence in comparison to PSO and GA, but the population size needed in DE was higher than the other two methods.

## **Author details**

Bruno Augusto Angélico1, Márcio Mendonça1, Lúcia Valéria R. de Arruda2, Taufik Abrão3

1 Federal University of Technology - Paraná (UTFPR), Campus Cornélio Procópio, PR, Brazil 2 Federal University of Technology - Paraná (UTFPR), Campus Curitiba, PR, Brazil

3 State University of Londrina - Paraná (UEL), Londrina, PR, Brazil

## **References**

18 Search Algorithms

Number of Failures: 30 Probability of Success: 0.9970

Number of Failures: 0 Probability of Success: 1.0

Number of Failures: 897 Probability of Success: 0.9103

Number of Failures: 0 Probability of Success: 1.0

Number of Failures: 0 Probability of Success: 1.0

**Author details**

**Table 7.** GA, PSO and DE simulation results for PROC2.

Bruno Augusto Angélico1, Márcio Mendonça1, Lúcia Valéria R. de Arruda2, Taufik Abrão3

**PSO with** P = 10 *A*max 0.7040 0.6590 0.9029 0.9407 0.8134 0.7234 0.6700 0.8153 *A*min 0.6400 0.4130 0.6590 0.6590 0.6590 0.6590 0.6590 0.6590 *A*average 0.6592 0.5006 0.7079 0.7221 0.6809 0.6593 0.6592 0.6850

**GA with** P = 10 *A*max 0.6800 0.5200 0.9038 0.9409 0.7870 0.6700 0.6700 0.8153 *A*min 0.6400 0.4800 0.6590 0.6590 0.6590 0.6590 0.6590 0.6590 *A*average 0.6596 0.5014 0.7500 0.7717 0.7055 0.6596 0.6596 0.7082

**DE with** P = 10 *A*max 0.7523 0.6591 0.9120 0.9477 0.8134 0.7608 0.7530 0.8271 *A*min 0.5747 0.4472 0.6590 0.6590 0.6590 0.6590 0.6590 0.6590 *A*average 0.6597 0.5046 0.7553 0.7808 0.7026 0.6632 0.6632 0.7097

**PSO with** P = 20 *A*max 0.6800 0.5200 0.9040 0.9411 0.7869 0.6700 0.6700 0.8154 *A*min 0.6400 0.4800 0.6590 0.6590 0.6590 0.6590 0.6590 0.6590 *A*average 0.6594 0.5002 0.7116 0.7267 0.6813 0.6594 0.6593 0.6865

**DE with** P = 20 *A*max 0.6800 0.5200 0.9043 0.9427 0.7870 0.6700 0.6700 0.8154 *A*min 0.6400 0.4800 0.6590 0.6590 0.6590 0.6590 0.6590 0.6590 *A*average 0.6595 0.5003 0.7548 0.7827 0.7015 0.6608 0.6608 0.7124

In most scenarios, DE has presented a faster convergence in comparison to PSO and GA, but

1 Federal University of Technology - Paraná (UTFPR), Campus Cornélio Procópio, PR, Brazil

2 Federal University of Technology - Paraná (UTFPR), Campus Curitiba, PR, Brazil

3 State University of Londrina - Paraná (UEL), Londrina, PR, Brazil

the population size needed in DE was higher than the other two methods.


[15] Parsopoulos K.E. and Groumpos P.P. A first study of fuzzy cognitive maps learning using particle swarm optimization. In *IEEE 2003 Congress on Evolutionary Computation*, pages 1440–1447, 2003.

**Chapter 10**

(*i* = *1*, *2*, …, *n*) in series

there are various versions, which are proposed by the

**Optimal Allocation of Reliability in Series Parallel**

One of the most important problems in many industrial applications is the redundancy optimization problem. This latter is well known combinatorial optimization problem where the design goal is achieved by discrete choices made from elements available on the market. The natural objective function is to find the minimal cost configuration of a series-parallel system under availability constraints. The system is considered to have a range of performance levels from perfect working to total failure. In this case the system is called a *multi-state* system

suppliers on the market. Elements are characterized by their cost, performance and availability according to their version. For example, these elements can represent machines in a manufac‐ turing system to accomplish a task on product in our case they represent the whole of electrical power system (generating units, transformers and electric carrying lines devices). Each

elements may be chosen for any given system component. Each component can contain

A limitation can be undesirable or even unacceptable, where only identical elements are used in parallel (i.e. homogeneous system) for two reasons. First, by allowing different versions of the devices to be allocated in the same system, one can obtain a solution that provides the desired availability or reliability level with a lower cost than in the solution with identical parallel devices. Second, in practice the designer often has to include additional devices in the existing system. It may be necessary, for example, to modernize a production line system according to a new demand

contains a number of elements connected in parallel. Different versions of

© 2013 Abdelkader et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2013 Abdelkader et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

(MSS). Let consider a multi-state system containing *n* components *Ci*

levels from customers or according to new reliability requirements.

**Production System**

http://dx.doi.org/10.5772/55725

arrangement. For each component *Ci*

elements of *different* versions as sketched in figure 1.

**1. Introduction**

component *Ci*

Rami Abdelkader, Zeblah Abdelkader, Rahli Mustapha and Massim Yamani

Additional information is available at the end of the chapter

