Optimal location and setting of SVC and TCSC devices using non-dominated sorting particle swarm optimization

Electric Power Systems Research 79 (2009) 1668–1677

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Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr

Optimal location and setting of SVC and TCSC devices using non-dominated sorting particle swarm optimization R. Benabid a,∗ , M. Boudour b , M.A. Abido c a b c

Nuclear Research Center of Birine, B.P. 180, 17200 Ain oussera, Djelfa, Algeria Department of Electrical Engineering University of Sciences & Technology Houari Boumediene (U.S.T.H.B), El Alia, BP 32, Bab Ezzouar, 16111 Algiers, Algeria Electrical Engineering Department, King Fahd University of Petroleum and Mineral, Box 1225, Dhahran 183, Saudi Arabia

a r t i c l e

i n f o

Article history: Received 17 May 2008 Received in revised form 19 March 2009 Accepted 13 July 2009 Available online 25 August 2009 Keywords: Static voltage stability margin Multi-objective optimization Particle swarm optimization Non-dominated sorting particle swarm optimization Non-dominated sorting genetic algorithms II SVC TCSC

a b s t r a c t In this paper, a new method for optimal locating multi-type FACTS devices in order to optimize multiobjective voltage stability problem is presented. The proposed methodology is based on a new variant of particle swarm optimization (PSO) specialized in multi-objective optimization problem known as nondominated sorting particle swarm optimization (NSPSO). The crowding distance technique is used to maintain the Pareto front size at the chosen limit, without destroying its characteristics. To aid the decision maker choosing the best compromise solution from the Pareto front, the fuzzy-based mechanism is employed for this task. NSPSO is used to find the optimal location and setting of two types of FACTS namely: Thyristor controlled series compensator (TCSC) and static var compensator (SVC) that maximize static voltage stability margin (SVSM), reduce real power losses (RPL), and load voltage deviation (LVD). The optimization is carried out on two and three objective functions for various FACTS combinations considering. For ensure the robustness of the proposed method and gives a practical sense of our study, N − 1 contingency analysis and the stress of power system is considered in the optimization process. The thermal limits of lines and voltage limits of load buses are considered as the security constraints. The proposed method is validated on IEEE 30-bus and realistic Algerian 114-bus power system. The simulation results are compared with those obtained by particle swarm optimization (PSO) and non-dominated sorting genetic algorithms (NSGA-II). The comparisons show the effectiveness of the proposed NSPSO to solve the multi-objective optimization problem and capture Pareto optimal solutions with satisfactory diversity characteristics. © 2009 Elsevier B.V. All rights reserved.

1. Introduction In the last few years, voltage collapse problems in power systems have been of permanent concern for electric utilities: several major blackouts throughout the world have been directly associated to this phenomenon, e.g. in France, Italy, Japan, Great Britain, WSCC in USA, etc. [1]. The voltage instability can occur when a power system is heavily loaded in transmission lines and/or lacks in local reactive power sources [2]. Several efforts have been made to find the ways to ensure the security of the system in terms of voltage stability. It is found that flexible alternative current transmission system (FACTS) devices are a good choice to improve the static voltage stability margin (SVSM) in power systems, which operates near the steady-state stability limit and may result in voltage instability [3]. Moreover it can provide benefits in increasing system transmission capacity and power

∗ Corresponding author. Tel.: +213 775172313. E-mail address: rabah [email protected] (R. Benabid). 0378-7796/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2009.07.004

flow control flexibility and rapidity [4]. Taking advantages of the FACTS devices depends greatly on how these devices are placed in the power system, namely on their location and size [5]. In a practical power system, allocation of FACTS devices depends on a comprehensive analysis of steady-state stability, small signal stability, voltage stability, and other practical factors such as cost and installation conditions also need to be considered [3]. In the literature, the optimal location and setting of FACTS devices has retained the interest of worldwide researchers in power systems, where various methods and criteria are used in this field. The optimal location of SVC and other types of shunt compensation devices for voltage stability enhancement is suggested in [6]. The authors used the modal analysis method in the vicinity of the point of collapse of the system test. In [7] an approach combining static and dynamic procedure is proposed to optimize the locations, types and ratings of two kinds of FACTS, namely, the SVC and TCSC to increase asset utilization of power systems. The procedure is based on the use of a continuation power flow, an optimal power flow, and an eigenvalue analysis. In [8], a sensitivity-based approach has been developed for determining the optimal placement of FACTS devices

R. Benabid et al. / Electric Power Systems Research 79 (2009) 1668–1677

in an electricity market having pool and contractual dispatches. In paper [9], the authors have also, presented a new sensitivity based approach for the optimal placement of the SVC to simultaneously enhance the static and dynamic voltage security of the power system. In [10], the authors investigated the problem of optimal number and location of TCSC in the deregulated electricity markets using mixed integer non-linear programming approach. Rather than the mathematical methods cited above, population based heuristic methods have been applied to the problem of optimal location of FACTS devices in power systems. Farsangi et al. [5] used particle swarm optimization (PSO) and genetic algorithms (GA) for SVC planning in order to enhance voltage profile and to reduce total real power losses. The two objectives were considered as the inputs of the fuzzy inference system and the output was an index of satisfaction of objectives. In [4] the PSO technique was used to find the optimal location of multi-type of FACTS devices, namely SVC, TCSC, and UPFC with minimum cost of installation and to improve the system loadability. The two objectives were converted into a single objective function. In [11] GA was used to optimize four type of FACTS devices namely TCSC, TCVR, TCPST, SVC, in order to enhance the 118-bus power system loadability. The optimizations were performed on three parameters: locations, types, and settings of these devices. In [12], the authors applied the Genetic Algorithm to find the optimal types and ratings of UPFC, TCSC, TCPST, and SVC that ensure the minimal overall system cost function. The later includes the investment costs of the FACTS devices and the bid offers of the market participants. The two objectives were transformed into a single objective function. In [13], a genetic algorithm-based method was used to identify the optimal number and location of UPFC devices in an assigned power system network for maximizing system capabilities, social welfare and to satisfy contractual requirements in an open market power. The compromise between all these objective functions was calculated using fuzzy logic. From the previous works, we can conclude that the problem of optimal location of FACTS devices is generally formulated as a mono-objective optimization problem that optimize a single objective function, transform several objectives to a single objective by aggregating them, or via a fuzzy inference system. The formulation of optimal location of FACTS as multi-objective optimization problem is a new attempt in this field, the authors in [14], use a multi-objective particle swarm optimization (MOPSO) Algorithm to find the optimal location of thyristor controlled series compensator (TCSC) and its parameters in order to increase the total transfer capability (TTC), to reduce total transmission losses and to minimize voltage deviation. This paper investigates the optimal location of FACTS devices as a real multi-objective optimization problem. In doing so, we use a non-dominated sorting particle swarm optimization (NSPSO) method to find the optimal setting and placement of the two popular FACTS namely: TCSC and SVC considering different objectives such as increasing SVSM, minimizing real power losses (RPL), and minimizing load voltage deviation (LVD). The optimization procedure is performed for two and three objective functions. Firstly, the problem is formulated as a bi-objective optimization problem, considering only the minimization of RPL and the maximization of SVSM. In the second step, three objectives are optimized, considering also, the minimization of load LVD. This paper is organized as follows: Section 2 presents a brief introduction of multi-objective optimization problems. In Section 3 the NSPSO algorithm is presented along with a detailed discussion. The FACTS modeling and problem formulation are presented in Section 4. The results and discussion are presented in Section 5. Finally, major contributions and conclusions are summarized in Section 6.

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2. Non-dominated sorting particle swarm optimization method 2.1. Multi-objective optimization overview Many optimization problems in the world involve the optimization of several objectives at the same time. To obtain the optimal solution, there will be a set of optimal trade-offs between the conflicting objectives, where the set of optimal solution is known as Pareto front [15]. A multi-objective optimization problem is defined as the maximization or the minimization of many objectives subject to equality and inequality constraints. The multi-objective optimization problem can be formulated as follows: Minimize fi (x),

i = 1, . . . , Nobj

Subject to constraints :



gj (x) = 0, hk (x) ≤ 0,

(1) j = 1, ..., M k = 1, ..., K

(2)

where fi is the ith objective function, x is the decision vector, Nobj is the number of objectives, gj is the jth equality constraint, and hk is the kth inequality constraint. Several papers proposed to extend the particle swarm optimization (PSO) method in order to handle a multi-objective optimization problem [16–22]. Among these algorithms, NSPSO algorithm is based on the same non-dominated sorting concept used in NSGA-II [22]. This approach will ensure more non-dominated solutions discovered through the domination comparison operations. NSPSO is presented in detail bellow. 2.2. Non-dominated sorting particle swarm optimization principle Instead of comparing solely on a particle’s personal best with its potential offspring, the entire population of N particles’ personal bests and N of these particles’ offspring are first combined to form a temporary population of 2N particles. After this, the nondominated sorting concept is applied, where the entire population is sorted into various non-domination fronts. The first front being completely a non-dominant set in the current population and the second front being dominated by the individuals in the first front only and the front goes so on. Each individual in each front is assigned fitness values or based on front in which they belong to. Individuals in the first front are given a fitness value of 1 and individuals in second are assigned a fitness value of 2 and so on. In addition to the fitness value, a new parameter called crowding distance is calculated for each individual to ensure the best distribution of the non-dominated solutions. The crowding distance is a measure of how close an individual is to its neighbors. The global best gbesti for the ith particle xi is selected randomly from the top part of the first front (the particle which has the highest crowding distance). N particles are selected based on fitness and the crowding distance to play the role of pbest. Such as, when the first front has more than N particles, we select the particles that have the highest distance, the principle of selection of pbest is well presented in Fig. 1. The update of the particles position in the research space is based on the two following equations [23]: xik+1 = xik + vk+1 i

(3)

vk+1 = wvki + c1 rand1 × (pbesti − xik ) + c2 rand2 × (gbesti − xik ) i

(4)

where w is the weighting function, cj the weighting factor, rand the random number between 0 and 1, pbesti the personal best of the particle I, gbesti the global best of the particle i, vki the current

the current velocity of agent i velocity of agent i at iteration k, vk+1 i at iteration k + 1, xik the current position of agent i at iteration k, and

xik+1 the current position of agent i at iteration k + 1.

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(choose the first 7. |pbestk+1 | = |pbestk+1 | ∪ Fi (N − |pbestk+1 |) N − |pbest| elements of Fi ); 8. xk+1 (use (4) and (5) to calculate the new positions of particle with using the new pbest and gbest). 䊉 k = k + 1. 2.3. Reducing Pareto set Generally, the Pareto-optimal set can be extremely large in some problems. In this case, reducing the set of non-dominated solutions without destroying the characteristic of the Pareto front is desirable from the decision maker’s point of view [15]. In this work, if the Pareto front set exceeds its limit preset by the user, we remove the solutions that have the smallest Crowding distance. This technique ensures a well distribution of non-dominated solutions in the Pareto front.

Fig. 1. Procedure of selection of pbest.

The following weighting function is used in this study [23]: w = wmax −

wmax − wmin × iter itermax

(5)

where wmax the initial weight, wmin the final weight, itermax the maximum iteration number, and iter the current iteration number. The steps of basic NSPSO algorithm are depicted in Fig. 2 and summarized below. For each iteration k do: 1. Rk = xk ∪ pbestk (combine the current solution and all personal best); 2. F = non dom sort(Rt ) (application the non-dominated sorting on Rt ); 3. pbestk+1 =  and i = 1; 4. until |pbestk+1 | + |Fi | ≤ N (until the pbest set is filled); a. i = i + 1; b. calculate the crowding distance for each particle in Fi ; c. pbestk+1 = pbestk+1 ∪ Fi ; 5. sort (Fi ) (sort in descending order); 6. select randomly gbest for each particle from a specified top part (e.g. top 5%) of the first front F1 ;

2.4. Best compromise solution Once the Pareto optimal set is obtained, it is practical to choose one solution from all solutions that satisfy different goals to some extends [25]. Due to the imprecise nature of the decision maker’s (DM) judgment, it is natural to assume that the DM may have fuzzy or imprecise nature goals of each objective function [26]. Hence, the membership functions are introduced to represents the goals of each objective function; each membership function is defined by the experiences and intuitive knowledge of the decision maker [27]. In this study, a simple linear membership function was considered for each of the objective functions. The membership function is defined as follows [26]:

i =

⎧ 1, ⎪ ⎨ F max − F i

i

max min ⎪ ⎩ Fi − Fi

0,

Fi ≥ Fimax , ,

Fimin < Fi < Fimax , Fi ≤ Fimin

Fig. 2. Non-dominated sorting particle swarm optimization algorithm.

(6)

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presented in Fig. 4a. The basic idea behind power flow control with the TCSC is to decrease or increase the overall lines effective series transmission impedance, by adding a capacitive or inductive reactive correspondingly. The TCSC is modeled as variable impedance as depicted in Fig. 4b. After installing TCSC, the new reactance of line is presented by Eq. (9). Xij = (1 + k)X

Fig. 3. Static var compensator presentation: (a) basic structure, (b) model

where Fimin and Fimax are the minimum and the maximum value of the ith objective function among all non-dominated solutions, respectively. The membership function  is varied between 0 and 1, where  = 0 indicates the incompatibility of the solution with the set, while  = 1 means full compatibility [26]. For each non-dominated solution k, the normalized membership function k is calculated as

Nobj

k =

i=1

k

M Nobji k=1

i=1

ki

(7)

where M is the number of non-dominated solutions and Nobj is the number of objective functions. The function k can be considered as a membership function of non-dominated solutions in a fuzzy set, where the solution having the maximum membership in the fuzzy set is considered as the best compromise solution. 3. FACTS modeling As we already mentioned, this paper focuses on the optimal location and settings of two kinds of FACTS, namely the SVC and the TCSC. The model of these FACTS used in this paper is presented in detail bellow.

(9)

where X is the transmission line reactance and k is the level of reactance compensation. The level of the applied compensation of the TCSC varies generally between 20% inductive and 80% capacitive [11].

4. Problem formulation The optimal location and settings of SVC and TCSC is formulated as a real constrained mixed discrete-continuous multi-objective optimization problem. Firstly, the optimization problem is formulated as bi-objective optimization problem considering static voltage stability margin and power losses objectives. Second, the problem is formulated as a three-objective optimization problem, where we added the voltage deviation in load buses to the first problem. In all optimization problems several cases in terms of use of FACTS devices are considered namely (1) Case 1: SVC only, (2) Case 2: TCSC only, (3) Case 3: coordinated SVC and TCSC. The objective functions considered in this paper are presented in detail below.

3.1. Model of SVC The SVC is a shunt compensator that may have two characters: inductive or capacitive. In the first case, it absorbs reactive power while in the second one the reactive power is injected [11]. The SVC combines a series capacitor bank shunted by Thyristor controlled reactor (see Fig. 3a). In this paper, the SVC is modeled as a variable admittance like is presented in Fig. 3b. The reactive power provided is limited as presented in Eq. (8). min bmin SVC ≤ bSVC ≤ bSVC

(8)

3.2. Model of TCSC

4.1. Static voltage stability margin (SVSM) Static voltage stability margin (SVSM) or loading margin is the most widely accepted index for proximity of voltage collapse. SVSM is defined as the largest load change that the power system may sustain at a bus or collective of buses from a well defined operating point (base case). The SVSM is calculated considering FACTS devices using power system analysis toolbox (PSAT) [24]. The maximization of SVSM can be presented as follows: Max{}

The TCSC is a series compensation component which consists of a series capacitor bank shunted by Thyristor controlled reactor as

where  is the SVSM or the loading margin.

Fig. 4. Thyristor controlled series compensator presentation: (a) basic structure, (b) model.

(10)

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4.2. Real power losses (RPL) This objective consists of minimizing the real power loss in the transmission lines and which can be expressed as

Min

 nl

gk [Vi2

+ Vi2

− 2Vi Vj cos(ıi − ıi )]

(11)

After the evaluation of the objective functions and the checking of the inequality constraints of each particle in NSPSO, we compute its corresponding SVSM considering N − 1 contingency and power system stress. If there is a voltage collapse, the current solution is rejected by assigning a bad fitness. Consequently, this approach will ensure an effective optimization of FACTS devices and makes sure the voltage security of power system.

k=1

where nl is the number of transmission lines; gk is the conductance of the kth line; Vi < ıi and Vj < ıj are the voltages at the end buses i and j of the kth line, respectively. 4.3. Load voltage deviation (LVD) This function is to minimize the deviations in voltage magnitudes at load buses that can be expressed as Min

 NL



5. Results and discussions In order to investigates the effectiveness of the proposed approach. NSPSO was tested on IEEE 30-bus six generators [28], and the realistic Algerian 114-bus power system [29]. The generators are modeled as PV buses with Q limits; the loads are typically represented by constant PQ loads; where the increase in the load is regarded as the parameter which leads the power system to a voltage collapse.



ref |Vk − Vk |

(12)

k=1 ref

where NL is the number of load buses and Vk

is the pre-specified

ref reference value of the voltage magnitude at the kth load bus. Vk

is

usually set to 1.0 pu.

4.4. Equality and inequality constraints The equality and inequality constraints must be satisfied during the optimization procedure. The equality constraints represent the typical load flow equations. The inequality constraints represent the reactive power limit of generators, and the operating limits of the TCSC and SVC. Moreover, two security limits are considered in this paper, namely the thermal limits of the transmission lines and the bus voltage limits, which are applied on the two last objectives only (RPL and LVD). For the SVSM objective, the security limits are not considered, because the voltage collapse occurs generally after the violation of the security limits. In this work, if the inequality limits are not satisfactory the current solution will be assigned by a bad fitness. 4.5. Including N − 1 contingency and power system stress in the optimal location of FACTS The power system is usually stressed by means of several causes such as the variation of load and/or outage of one component or more like lines or generators. In this paper and in the order to ensure an effective optimization of the SVC and the TCSC; we include the concept N − 1 contingency analysis and the variation of operation point in the optimal location of FACTS devices. However, the inclusion of all contingencies in an optimization problem is not practical and also time consuming. For overcome this problem; we perform an off-line N − 1 contingency analysis considering the outage of lines, transformers or generators, and we select the worst contingency that have the minimum SVSM. This contingency will be included in the optimization problem. Furthermore, the stress of power system is modeled by a variation of the system generation and load with 10% of the base case. The inclusion of N − 1 contingency and the stress of power system can be summarized as follows.

PL = P0L QL = Q0L

(13)

where  is loading margin that represents the SVSM, P0L and Q0L are the active and reactive base loads, whereas PL and QL are the active and reactive loads at a bus L for the current operating point. The load power factor is maintained constant during the load increasing. The decision variables considered, are the location and setting of TCSC and SVC. The number of FACTS and their constraints are chosen at the beginning; where the number of FACTS is fixed at one for each type, otherwise the reactance of TCSC is considered as continuous variable which varies between 20% inductive and 80% capacitive of the line reactance. The placement of TCSC is considered as a discreet variable, where all the lines of the system are selected to be the optimal location of TCSC. Similarly, the SVC considered as a generator (or an absorber) of reactive power which varies continuously between −2 pu and 2 pu. The optimal location of SVC is, also, considered as a discreet decision variable, where all load buses are selected to be the optimal location of SVC. From Pareto dominance viewpoint, the extreme points of the Pareto front present the optimal solution of each objective optimized individually. As a matter of fact, in order to evaluate the diversity characteristic of the obtained solutions, the extreme points of Pareto front are compared with the optimal solution of the correspondent objective optimized using particle swarm optimization (PSO). Furthermore, the results of NSPSO of the biobjective optimization problem are compared with those of the non-dominated sorting genetic algorithms (NSGA-II). 5.1. Static voltage stability margin and real power losses optimization 5.1.1. IEEE 30-bus The detailed data of IEEE 30-bus six generators are taken from [28]. The system has six generators located at buses 1, 2, 5, 8, and 13 and four transformers with off-nominal tap ratio in line 6–9, 6–10, 4–12, and 27–28. The lower voltage magnitude limits at all buses are 0.95 pu for all buses and the upper limits are 1.1 pu for generators 2, 5, 8, and 13, and 1.05 pu for the remaining buses including the reference bus 1. Moreover, the apparent power limits of lines are considered. The total active and reactive powers of system load are respectively: 283.4 MW, and 126.2 MVAR.

Table 1 NSPSO parameters. cj

wmax

wmin

Number of generation

Population size

Limit of Pareto front

2.0

0.9

0.4

100

100

30

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Fig. 7. Pareto front of case 2 for IEEE 30-bus test system. Fig. 5. Pareto front of case 1 for IEEE 30-bus test system.

Fig. 8. NSPSO and NSGA-II comparison of case 2 for IEEE 30-bus test system.

Fig. 6. NSPSO and NSGA-II comparison of case 1 for IEEE 30-bus test system.

During the optimization process, two scenarios are considered as follows: (a) Outage of line 27–29, (b) Stress of the system by 10% of the operating point. The parameters of NSPSO for all optimization cases are summarized in Table 1. 5.1.1.1. Case 1: SVC only. Fig. 5 depicts the Pareto-optimal set of case 1. It can be seen that the obtained solutions are well distributed on trade-off surface, except some discontinuity, caused by the discrete decision variables. Fig. 6 presents the Pareto front of the proposed method and NSGA-II. From this figure it can be seen that the NSPSO

is more efficient than NSGA-II in the viewpoint optimality and distribution of solutions in the trade-off surface. The best solution of SVSM and RPL optimized individually the PSO and the solutions obtained by the proposed method are presented in Table 2. From Table 2, it can be seen that the extreme points of the Pareto front and the best solution of each function are almost identical. So it can be concluded that the proposed method is capable of exploring more efficient the research space. Furthermore, we can conclude that the placement of SVC at bus 22 with the reference set at −1.05 pu provides the best SVSM of 3.024 pu. The installation of SVC at bus 21 with −1.043 pu of reference provides the minimum RPL of 0.0522 pu. From the decision maker’s point of view, the installation of SVC at bus 21 with −1.045 pu is considered as the best compromise solution in the overall non-dominated solutions set.

Table 2 NSPSO solutions of case 1 for bi-objective optimization. PSO

Location of SVC Setting of SVC SVSM RPL

NSPSO

Best SVSM

Best RPL

Best SVSM

Best RPL

Best compromise solution

22 −1.05 3.024 0.0526

21 −1.043 2.974 0.0522

22 −1.05 3.024 0.0526

21 −1.0429 2.97 0.05239

21 −1.045 2.997 0.0524

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optimal coordination between the SVC and TCSC. Table 4 presents the best solution of SVSM and RPL and NSPSO results of case 3. The comparison of these results shows that the proposed method is well explored the research space. Compared with the two previous cases, case 3 gives the best SVSM. 5.1.2. Algerian 114-bus power system In order to give a more practical aspect to our work, we applied the proposed method on the realistic Algerian power system 114-bus [29]. The system has 15 generators, 159 lines and 16 transformers with off-nominal tap ratio. The total active and reactive load of the system is 3727 MW and 2070 MVAR respectively. The lower and the upper voltage limits of all buses are 0.9 pu and 1.1 pu respectively. During the optimization process, two scenarios are considered as follows: (a) Outage of generator number 11 at bus “JIJEL”, (b) Stress of the system by 10% of the operating point. Fig. 9. Pareto front of case 3 for IEEE 30-bus test system.

Because there are several results, we optimize only the case 3 for the Algerian 114-bus power system. 5.1.1.2. Case 2: TCSC only. The optimal location and setting of TCSC is presented in Fig. 7. From Fig. 7, it can be concluded that the NSPSO provides well-distributed solutions over the trade-off surface. The Pareto front of NSPSO and NSGA-II are presented in Fig. 8. From this figure, we confirm the superiority of the proposed method NSPSO over NSGA-II in point of view the well distribution and optimality of the solutions. The optimal solution of each objective and the extreme points given by NSPSO are presented in Table 3. From the results depicted in Table 3, it can be concluded that the research space is well explored, in such a way that the extreme points of each objective is the optimal value provided by PSO. In this case, all the non-dominated solutions indicate that the TCSC is installed at the line number 36 with different settings. From Table 3, it can be concluded that the installation of a TCSC in line 36 with 0.8 pu of reference provides the best SVSM of 2.416 pu, and the installation of TSCS in line 36 with −0.24247 pu of reference provides the best RPL of 0.0529 pu. In decision maker point of view, the installation of TCSC at line 36 with 0.6215 pu of reference is considered as the best compromise solution in overall non-dominated solutions. 5.1.1.3. Case 3: coordinated SVC and TCSC. The Pareto optimal of case 3 is presented in Fig. 9. Actually, the obtained solutions present the

5.1.2.1. Case 3: coordinated SVC and TCSC. The Pareto front is depicted in Fig. 10. From this figure, we can conclude that the proposed method is able to solve the optimal location of FACTS devices in realistic power system. 5.2. Voltage stability margin, real power losses, and load voltage deviation optimization This optimization problem is more complex than the previous, because we optimize three objective functions at the same time. The optimization problem consists of the maximization of SVSM, minimization of RPL, and LVD for the three cases cited previously. 5.2.1. IEEE 30-bus 5.2.1.1. SVC only. Fig. 11 depicts the Pareto front of the optimization problem of case 1. The extreme points obtained by NSPSO, and the optimal solutions of the mono-objective optimization are summarized in Table 5. The comparison of the extreme points obtained by NSPSO and mono-objective optimization solutions indicates that the research space is well explored by the proposed approach. From Table 5, it is clear that the installation of SVC at bus 7 with −1.2145 pu of reference provides the best SVSM of 0.777 pu. The

Table 3 NSPSO solutions of case 2 for bi-objective optimization. PSO

Location of TCSC Setting of TCSC SVSM RPL

NSPSO

Best SVSM

Best RPL

Best SVSM

Best RPL

Best compromise solution

36 0.8 2.4160 0.0533

36 0.4247 2.3581 0.0529

36 0.8 2.4160 0.0533

36 0.4247 2.3581 0.0529

36 0.6215 2.3903 0.0530

Table 4 NSPSO solutions of case 3 for bi objective optimization. PSO

Location of SVC Setting of SVC Location of TCSC Setting of TCSC SVSM RPL

NSPSO

Best SVSM

Best RPL

Best SVSM

Best RPL

Best compromise solution

24 −1.1 2 0.8 3.2 0.0667

21 −1.044 1 −0.2 2.9819 0.523

24 −1.1 2 0.8 3.2 0.0667

21 −1.044 1 −0.2 2.982 0.523

22 −1.074 36 0.8 3.1777 0.0546

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Fig. 10. Pareto front of case 3 for the Algerian 114-bus power system. Fig. 12. Pareto front of case 2 for IEEE 30-bus test system

show that the research space is well explored. The obtained results presented in Table 6 indicate that the installation of a TCSC at line1 with −0.6265 pu of reference provides the best SVSM of 0.6564 pu. Also, the installation of a TCSC at line 5 with −0.2503 pu of reference provides the best RPL of 0.1743 pu. The comparison of the extreme points obtained for SVSM and RPL with those presented in Table 3 indicates that the extreme points are almost the same; so we conclude that the NSPSO algorithm is robust. From Table 6, we conclude that the installation of TCSC at line 1 with −0.6312 pu of reference provides the best LVD of 0.6005 pu. From the decision maker point of view, the best compromise solution is the same obtained for the best SVSM.

Fig. 11. Pareto front of case 1 for IEEE 30-bus test system.

installation of SVC at bus 4 with −0.3915 pu provides the best RPL of 0.174 pu. The extreme points are almost the same as obtained in the previous problem for case 1, which demonstrates the robustness of the proposed method. Furthermore, the installation of SVC at bus 15 with 0.2056 pu of reference provides the best LVD of 0.3781 pu. From all the non-dominated solutions, the installation of SVC at bus 9 with −0.2789 pu of reference is considered as the best compromise solution. 5.2.1.2. TCSC only. The Pareto front of this case is presented in Fig. 12, and the extreme points and the optimal solution of the mono-objective optimization are presented in Table 6. The comparison of the extreme point and the optimal solution of each objectives

5.2.1.3. Coordinated SVC and TCSC. The Pareto front of this case is presented in Fig. 13, and the extreme points and the optimal solution of the mono-objective optimization are given in Table 7. The comparisons of the extreme point and the optimal solution of each objective show that NSPSO explores well the research space. The obtained results presented in Table 7 indicates that the installation of a SVC at bus 4 with −2 pu of reference and a TCSC at line 41 with 0.0938 pu of reference provides the best SVSM of 1.0505 pu. Also, the installation of a SVC at bus 4 with −0.3731 pu of reference and a TCSC at line 5 with −0.2549 pu of reference provides the best RPL of 0.1730 pu. The comparison of the extreme points obtained for SVSM and RPL with those presented in Table 4 indicates that the extreme points are almost the same; so we conclude that the NSPSO algorithm is robust. From Table 7, we conclude that the installation of a SVC at bus 12 with 0.3502 pu of reference and a TCSC at line 15 with −0.1236 pu of reference provides the best LVD of 0.2461 pu. From the decision maker point of view, the best compromise solution is the same obtained for the best RPL.

Table 5 The best solution of the three functions optimized individually, and NSPSO solutions of case 1. PSO

Location of SVC Setting of SVC SVSM RPL LVD

NSPSO

Best SVSM

Best RPL

Best LVD

Best SVSM

Best RPL

Best LVD

Best compromise solution

22 −1.05 3.024 0.0526 0.6008

21 −1.043 2.974 0.0522 0.6288

1 −0.6311 0.6561 0.1839 0.6005

22 −1.05 3.024 0.0526 0.6008

21 −1.043 2.974 0.0522 0.6288

1 −0.6312 0.6561 0.1839 0.6005

21 −1.045 2.997 0.0524 0.61

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Table 6 The best solution of the three functions optimized individually, and NSPSO solutions of case 2. PSO

Location of TCSC Setting of TCSC SVSM RPL LVD

NSPSO

Best SVSM

Best RPL

Best LVD

Best SVSM

Best RPL

Best LVD

Best compromise solution

36 0.8 2.4160 0.0533 0.6008

36 0.4247 2.3581 0.0529 0.6288

1 −0.6311 0.6561 0.1839 0.6005

1 −0.6265 0.6564 0.1838 0.6008

36 0.8 2.4160 0.0533 0.6288

36 0.4247 2.3581 0.0529 0.6005

36 0.6215 2.3903 0.0530 0.616

Table 7 The best solution of the three functions optimized individually, and NSPSO solutions of case 3. PSO Best SVSM Location of SVC Setting of SVC Location of TCSC Setting of TCSC SVSM RPL LVD

24 −1.1 2 0.8 3.2 0.0667 1.4354

NSPSO Best RPL 21 −1.044 1 −0.2 2.9819 0.523 0.7244

Best LVD

Best SVSM

Best RPL

Best LVD

Best compromise solution

12 0.3554 11 −0.2268 0.3747 0.1791 0.2315

24 −1.1 2 0.8 3.2 0.0667 1.4372

21 −1.044 1 −0.2 2.9819 0.523 0.7397

12 0.3502 15 −0.1236 0.3858 0.1787 0.2461

22 −1.074 36 0.8 3.1777 0.0546 0.7397

5.2.2. Algerian 114-bus power system 5.2.2.1. Case 3: coordinated SVC and TCSC. The Pareto front of this case is depicted in Fig. 14. From this figure, we can conclude that the NSPSO is able to solve the optimal location of FACTS devices formulated as multi-objective optimization problem and applied to realistic power system. 6. Conclusion

Fig. 13. Pareto front of case 3 for IEEE 30-bus test system.

In this work, a novel approach based on NSPSO has been presented and applied to optimal location and setting of SVC and TCSC. The problem is formulated as a real mixed continuous-integer multi-objective optimization problem, where two different problems are considered. At first, two competing objectives namely: SVSM and RPL are considered. In the Second problem, three objectives are considered, where we added the LVD to the first problem. In each case, the optimal location and setting of FACTS devices is performed for several use of FACTS. A crowding distance technique is used to maintain the Pareto front size at the chosen limit; moreover, a fuzzy-based mechanism is employed to extract the best compromise solution from the Pareto front. The results show that NSPSO provides well-distributed non-dominated solutions and well exploration of the research space. Moreover the method does not impose any limitation on the number of objectives. This work will be further extended to address the problem of optimal location of FACTS devices to enhance dynamic voltage stability. Acknowledgements The authors would like to thank Prof. F. Milano for its free PSAT package, and Dr. S.P. Torres for his helps and advices. References

Fig. 14. Pareto front of case 3 for the Algerian 114-bus power system.

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