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Journal of Intelligent & Fuzzy Systems 26 (2014) 681–692 DOI:10.3233/IFS-120759 IOS Press
Modified shuffled frog leaping algorithm for multi-objective optimal power flow with FACTS devices Rasoul Azizipanah-Abarghooeea,∗ , Mohammad Rasoul Narimania , Bahman Bahmani-Firouzia and Taher Niknamb a Department b Department
of Electrical Engineering, Marvdasht Branch, Islamic Azad University, Marvdasht, Iran of Electronic and Electrical Engineering, Shiraz University of Technology, Shiraz, Iran
Abstract. This paper presents a novel approach to depict Flexible AC Transmission Systems (FACTS) devices effects in power system using multi-objective optimization function. The FACTS devices can play very important roles in power system such as improve power system security, reduce generation cost, decrease transmission loss and improve the voltage stability index. Two more common FACTS devices are the Thyristor Controlled Series Capacitor (TCSC) and Static VAR Compensator (SVC) which can smoothly and rapidly change their apparent reactance and injection power respectively according to the system requirements. Determining the FACTS devices parameters in power system is too complicate and has a lot of local optima in its search space. In order to overcome above problems a new method, based on SFLA algorithm combined with a new mutation is proposed to increase the efficiency of the SFLA algorithm. Since the proposed problem is a multi-objective problem it is usual to obtain a set of solution instead one solution therefore Pareto method that uses concept of non-dominate solutions is applied to find best compromise solutions. An external repository is considered for saving all non-dominated solution, and also they are sorted by fuzzy set rule to obtain best solutions. For more validation the simulation results are compared with those in other literatures. Keywords: FACTS devices, fuzzy set, multi-objective optimization, modified SFLA, non-dominated solution
1. Introduction With expanding power transmission systems operation and control of them become gradually more complicate, application of FACTS devices becomes an appropriate alternative for solving these problems. The FACTS devices improve the stability of power systems both with their fast control characteristics and continuous compensating capability. In addition, they can be a proper alternative to increase load ability, decrease transmission loss, improve stability of power networks ∗ Corresponding author. Rasoul Azizipanah-Abarghooee, Department of Electrical Engineering, Marvdasht Branch, Islamic Azad University, Marvdasht, Iran. Tel.: +98 913 254 6011; Fax: +98 711 7353502; E-mail:
[email protected].
and reduce generation cost. According to the characteristics of the FACTS devices, various objectives have been considered in literatures to determine optimal parameters of these devices. Some of these reported objectives are voltage stability enhancement [1], diminish installation cost of FACTS devices [2], network security enhancement [3], enhancement power system load ability [4], increase power systems stability [5–7] and generation fuel cost reduction [8–10]. Nowadays due to rise of fuel prices and consequently rise in electricity power generation cost, it is crucial to find a way in order to generate electricity power with minimum cost while satisfying certain specified physical and operating constraints. The Optimal Power Flow (OPF) can satisfy above condition so that’s why it is considered as
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an important problem in power system sector [11, 12]. Also for a secure operation of the power system, it is important to maintain required level of stability. Therefore, the voltage stability enhancement index related to bus voltage magnitudes is considered as objective function and optimized in this paper to increase the secure operation of power system. Furthermore with expansion of power network and increase of power electricity demand, the transmission loss in power system becomes an important problem, because during the power transmission procedure from generation site to consumption one a lot of energy is lost in transmission lines, hence it is important to keep the transmission loss in acceptable range or reduce it. For above conditions loss and voltage stability enhancement index are considered as objective functions beside generation cost. For achieving to above condition we should solve a multi-objective problem which optimizes three objectives simultaneously. The basic target is to find proper adjustments of controlling variables, such as generator voltages and output power, transformer taps, shunt capacitors, TCSC and SVC parameters that would maintain acceptable cost, loss and voltage stability index. One important factor in application of FACTS devices in power systems is defining a proper model for them. In this regard, several TCSC models have been developed and used by the researchers depending on the applications [13–16]. In this paper, TCSC and SVC devices are modeled as injecting power devices which injects certain amount of power in special site that they are located, these models are simple and proper alternative for FACTS devices. As mentioned before adjust parameters of FACTS devices is a non-linear and complicate problem due to the non-linearity equation of FACTS devices which needs to solve with strong and accurate algorithm. Shuffled Frog Leaping Algorithm (SFLA) is a powerful memetic meta-heuristic algorithm proposed by Eusuff and Lansey in 2003 [17]. SFLA is a very simple, robust and fast algorithm, also it has been applied to solve complicate optimization problem [18, 19] successfully. However, the original SFLA often suffers problems such as trapped in local optima or converge to optimal value in long time. In order to avoid these problems, a new mutation is utilized, with modifying SFLA it called MSFLA and can overcome to above problems. Multi-objective optimization is a methodology that optimizes a group of objective functions simultaneously, it has been more and more used for optimization with multiple contradictory objectives. One traditional algorithm to solve multi objective problems is to combine all objective functions into a single one [20], the
other traditional algorithm is to select only one objective and join other objectives as constraints. There are some imperfections associated with these algorithms for example limit the available choices and need a priori selection of weights or targets for each objective function. As mentioned the aim of this paper is optimized three objectives function simultaneously, in this regard we use Pareto optimization method that benefit from non-dominate solution concept. The best privilege of Pareto method is to obtain a squad solution results in multi-objective optimization problems. There are a lot of combinations of FACTS parameters to achieve optimal solution and that’s why we use Pareto method in this paper. Also FACS devices cost depends on their parameters, with owning varieties combination of these parameters we can choose economically and particularly ones. The simulation results show that the proposed method is capable of obtaining accurate and acceptable solutions, also to validate the obtained results, they are compared with original SFLA, genetic algorithm (GA) and the particle swarm optimization (PSO) methods. The remainder of this paper is organized as follows. In Sections 2 and 3, the steady state model of SVC and TCSC are proposed, respectively. In Section 4, the proposed OPF problem considering FACTS devices is formulated. The multi-objective solution methodology is presented in Section 5. In Section 6, the new MSFLA algorithm is described. The application of MSFLA algorithm to multi objective problem is presented in Section 7. The feasibility and efficiency of the proposed method is investigated on IEEE 30-bus test system. Finally, in Section 9, the relevant conclusions are extracted.
2. Steady state model of SVC Static VAR Compensator (SVC) comprises a group of shunt connected capacitors and reactors banks with fast control action by means of thyristor switching. From the operational point of view, SVC can be seen as a variable shunt reactance that is adjusted automatically in response to change system operative conditions. Depending on the nature of the equivalent SVC’s reactance, the SVC draws either capacitive or inductive current from the network. A simple SVC that works both in the capacitive and inductive range can be obtained by a Fixed Capacitor (FC) in parallel with a Thyristor Controlled Reactor (TCR). Therefore, behavior of SVC seems as shunt-connected variable
R. Azizipanah-Abarghooee et al. / Modified shuffled frog leaping algorithm for multi-objective optimal power flow
Fig. 2. One line diagram of TCSC in line.
Fig. 1. One phase diagram of Static VAR Compensator (SVC).
reactance, which either generates or absorbs reactive power in order to regulate the voltage magnitude at the point of connection to the AC network. One-phase diagram of SVC is depicted in Fig. 1. The reactive power injected or absorbed by the SVC is used as the state variable of this model (QSVC ). The operational technical limit for this variable is Qmin ≤ QSVC ≤ Qmax .
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Where, Gsr =
XTCSC Rsr (XTCSC − 2Xsr ) 2 )(R2 + (X − X 2 (R2sr + Xsr sr TCSC ) ) sr
Bsr = −
(5)
XTCSC Rsr (XTCSC − 2Xsr ) (6) 2 )(R2 + (X − X 2 (R2sr + Xsr sr TCSC ) ) sr
Xsr and Rsr are the admittance and resistant of the line between bus #s and bus #r, respectively.
3. Steady state model of TCSC 4. Problem formulation TCSC is a traditional FACTS device which can change power system parameters smoothly, there are two possible characteristics for TCSCs, capacitive and inductive, to increase or diminish the transmission line reactance. Figure 2 shows a model of transmission line that TCSC connected between buses S and R. During steady state, TCSC can be considered as a static reactance (−jXTCSC ). This controllable reactance XTCSC is directly used as the control variable in the power flow equations. The operational technical limit for this variable is |XTCSC | ≤ X TCSC max . The power injected equations of TCSC can be derived as follow [18]: Gsr cos(δs − δR ) 2 (1) Psr = Vs Gsr − Vs VR +Bsr sin(δs − δR ) Qsr =
−Vs2 Bsr
− V s VR
Prs = VR2 Gsr − Vs VR
−Bsr cos(δs − δR ) Gsr cos(δs − δR )
Qrs =
+ V s VR
(2)
(3)
−Bsr sin(δs − δR )
−VR2 Bsr
Gsr sin(δs − δR )
Gsr sin(δs − δR )
4.1. Cost The aim of this section is define the fuel cost minimization while satisfying operational constraints, basically the problem is formulated as follows [10]: FCost =
Ng
2 (ai PGi + bi PGi + ci ) $ h
(7)
i=1
Where FCost is a total fuel cost ($/hr), PGi is a active power generation of unit i (MW); ai , bi and ci are the fuel cost coefficients of unit I and Ng is a number of generators. 4.2. Loss Second objective which consider in this paper is power loss (Floss ). The power flow solution gives all bus voltage magnitudes and angles, then, the real power loss in transmission line can be computed as follow [21]: Floss =
NL
gk [Vi2 − Vj2 − 2Vi Vj cos(θi − θj )] (8)
j=1
+Bsr cos(δs − δR ) (4)
Where NL is the number of transmission lines, δi and δj are the bus voltage angle at the two end of kth line, Vi and Vj are bus voltage amplitudes at two end of kth line and gk is the conductance of the kth line, respectively.
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4.3. Voltage stability index Kessel and Glavitsch [22] proved that the static voltage stability index can be measured as follow: Ng V i Lj = 1 − Fji j = Ng + 1, . . . , n (9) Vj i=1
Where Ng is the number of generator buses, n is the total number of buses in the system. Matrix [F] is computed as follow [23]: [F ] = −[YLL ]−1 [YLG ]
(10)
Where [YLL ] and [YLG ] are sub matrices of Y bus matrix. The network equations in terms of the node admittance matrix can be simply written as: Ibus = Ybus Vbus
(11)
Segregating the nodes into two categories, that is, the load bus set αL and generator bus set αG , can be written as follow: IL YLL YLG VL = (12) IG YGL YGG VG The L indices for the given load condition are computed for all load buses and the maximum of L-indices gives the proximity of the system to voltage collapse. For stable situations the condition 0 ≤ Lj ≤ 1 must not be violated for any of the nodes j. Hence, a global indicator L describing the stability of the whole system is given by:
Where i = 1, 2, . . . , n and θij = θi − θj , that θi and θj are the voltage angle of two ending bus of an arbitrary branch, and n is a number of the bus. The OPF inequality constraints reflect the limits of physical devices in the power system as well as the limits created to ensure system security that they are presented in following inequalities [10]: Pgi min ≤ Pgi ≤ Pgi max ; i = 1, 2, . . . , Ng
(16)
Qgi min ≤ Qgi ≤ Qgi max ; i = 1, 2, . . . , Ng
(17)
Pij ≤ Pij max Vi min ≤ Vi ≤ Vi max ; i = 1, 2, . . . , Nl
(18) (19)
Where Nl is the number of load bus and Pij is the power that flows between bus i and bus j. Vi max and Vi min voltages are respectively maximum and minimum valid voltages for each bus. Pij max is the maximum power flows that can transfer through the branch. Pgi max and Pgi min are the maximum and minimum active power values of the ith bus respectively. Qgi max and Qgi min are maximum and minimum reactive power values of the ith bus. Also the operation constraint for the active and reactive powers for TCSC and SVC are: Qmin SVC ≤ QSVC ≤ Qmax SVC
(20)
|XTCSC | ≤ X TCSC max
(21)
5. Multi-objective solution strategy L = max(Lj )
j ∈ αL
(13)
4.4. Constraints The OPF equality constraints reflect the physics of the power system. Equality constraints are expressed in following equations [24]: Pi = Pgi − Pdi =
n
Vi Vj (Gij cos θij + Bij sin θij )
j=1
(14) Qi = Qgi − Qdi =
n
Vi Vj (Gij sin θij − Bij cos θij )
j=1
(15)
5.1. Fundamental of multi-objective optimization When we seek an optimal solution for a multiobjective problem, we expect finding one and only one solution. In fact, this is a perfect case, most of time we find a set of solutions, due to the contradictory objectives. Real world applications usually contain simultaneous optimization of multiple objectives, these problems have many solutions and the reason for the optimality of many solutions is that no one can be considered to be better than any other with toward all objective functions. A general multi-objective optimization problem comprises number of objective to be optimized simultaneously and is connected with a number of equality and
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inequality constraints. It can be formulated as follows [25]: Min fi (x) Subject to:
i = 1, 2, . . . Nobj
gj (x) = 0
j = 1, 2, . . . N
hk (x) ≤ 0
k = 1, 2, . . . E
(22)
Where fi is the ith objective function, x is a determination vector that presents a solution, Nobj is the number of objectives. N and E are the number of equality and inequality constraint, respectively. 5.2. Fuzzy modeling for normalized objective function In this section, a fuzzy method has been proposed to compute the normalized form of objective functions in regard to solve multi-objective optimization. Since the objective functions are imprecise and they are not in the same range, therefore they are formulated as fuzzy sets, a fuzzy set is generally shown by a membership function (µi ). The fuzzy set function is depicted by the membership function to substitute each objective function as a value between 0 and 1, as shown in Fig. 3. This decisions maker is fully satisfied with each of objective if µi = 1, and not satisfied at all if µi = 0. Therefore, the value of each membership function shows the adaptability of the related objective. 5.3. Pareto optimal solution Since in multi-objective optimization problems, objective functions are conflict each other and are not in the same range we have to normalize these objectives with fuzzy method which described in last section. Each one of three solutions µ1 , µ2 and µ3 can have one of two
Fig. 3. Membership functions for fuzzy decision making.
possibilities: one of them dominates the other or none dominates the other. A vector Fa is said to dominate another vector Fb , denote as [26]: Fa ≺ Fb , iff fa,i ≤ fb,i ∀i = {1, 2, . . . , M} and ∃j ∈ {1, 2, . . . , M} where fa,j ≺ fb,j
(23)
685
(24)
The solutions that are non-dominated during the entire search space are denoted as Pareto-optimal. In general, the target of a multi-objective optimization algorithm is not only guiding the search towards the Pareto-optimal front but also maintain squad of non-dominated population, therefore in the proposed algorithm we defined a repository to save all nondominated solutions in each iterate. The solutions that save in this repository in all iteration are sorted by a type of decision making. Sorted solutions are Pareto optimal set and we can elect the best solution set by select the top solutions in this assortment. Decision maker that utilized in this article, defined as follow [26]: n
Nµ (j) =
ωk × µjk
k=1 n m
(25) ωk × µjk
j=1 k=1
Where, ωk is the weight for the kth objective functions and m is the number of non-dominated solutions.
6. MSFLA algorithm 6.1. Overview of original SFLA SFLA is a decrease based stochastic search method that begins with an initial population of frogs that represent the decision variables. An initial population of frogs is created randomly, for k-dimensional problems (K variables), a frog i is represented as Xi = (xi1 , xi2 , . . . , xik ). Afterwards, the frogs are sorted in a descending order according to their fitness, after that total population is divided into groups (memeplexes) that search independently. In this process, the first frog goes to the first memeplex, the second frog goes to the second memeplex, frog m goes to the Mth memeplex, and (M + 1)th frog goes to the first memeplex, and so on. In each memeplex, the frogs with the best and the worst fitness are identified as Xb and Xw , respectively. Also, the frog with the best and worst fitness in all memeplexes are identified as Xg and Xw, respectively. Then, a process is applied to improve only the frog with the worst fitness in each iterate. Correspondingly, the
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location of the frog with the worst fitness is regulated as follows: Di = r and () × (Xb − Xw ); −Dmax ≤ Di ≤ Dmax (26) Xw(new) = Xw + Di
(27)
Where rand () is a random number between 0 and 1, and Dmax is the maximum permitted change in a frog’s location. If this process generates a better solution, it replaces the worst frog. Otherwise, the calculations in Equation (26) and (27) are repeated. In addition, to provide the opportunity for random generation of improved information, random virtual frogs are generated and substituted in the population if the local search cannot find better solutions respectively in each iterate. After a number of iterations, the different groups combined and share their ideas with themselves through a shuffling process. The local search and the shuffling processes continue until defined convergence criteria are satisfied. In this paper, a new strategy is proposed in order to increase the SFLA ability, this approach called Modified Shuffle Frog Leaping Algorithm (MSFLA) will be presented in details in following. 6.2. Modified Shuffle Frog Leaping Algorithm (MSFLA) For increasing the SFLA ability in finding the global optima a powerful mutation is used in this paper. The fundamental idea behind mutation is a scheme by which it generates the trial mutated vectors. These mutation operators can be described as follows: Mutation strategy 1: k k k k (.) Xmut,j,1 = Xm + r and Xg ; − Xw ,j 1 (28) j = 1, . . . , N1k
Based on a probability model described in [12], each frog selects one of these two methods.
7. Application of MSFLA algorithm to multi-objective problem This part demonstrates the application of the proposed algorithm for solving the multi-objective problem using non-dominated Pareto optimal method. Step 1: Define the input data. In this step, the input data including the generator real power, generator bus voltages, fuel cost coefficient of generators, emission coefficient of generators, tap of transformer, reactive power of switchable VAR, fimin and fimax are defined. Step 2: An initial population Xj which must meet constraints is generated randomly as follows: ⎤ ⎡ X1 ⎥ ⎢ ⎢ X2 ⎥ ⎥ ⎢ (30) Population = ⎢ ⎥ ⎢... ⎥ ⎦ ⎣ XNF Xi = [xi,1 , xi,2 , . . . , xi,N ] Xi = Vgi , Pgi , Qci , tap, TCSCparameter , SVCparameter
j = 1, . . . , N2k (29) Where, N1k and N2k are the respective number of frogs which choose the mutation strategy 1 and 2 in iterak k k tion k. Xm , Xm and Xm are three vectors that 1 ,j 2 ,j 3 ,j selected randomly between memeplexes in order to uniformly cover the algorithm search domain. All the frogs in the population will have a chance to mutate, controlled by the probability of their methods of mutating.
(32)
Where, Vgi = [Vgi,1 , Vgi,2 , . . . , Vgi,Ng ]1×Ng
(33)
Pgi = [Pgi,1 , Pgi,2 , . . . , Pgi,j , . . . Pgi,Ng−1 ]1×(Ng−1) ; j= / slack bus (34)
Mutation strategy 2:
k k k k (.) Xmut2,j = Xm + r and X ; − X ,j ,j ,j m m 1 2 3
(31)
Qci = [Qci,1 , Qci,2 , . . . , Qci,Nc ]1×Nc
(35)
TCSCparameter = [TCSCparameter ]1×1
(36)
SVCparameter = [SVCparameter ]1×1
(37)
xi,j = r and (.) ∗ xj,max − xj,min + xj,min j = 1, 2, 3, . . . , Nparam ; i = 1, 2, 3, . . . , NF
(38)
Where, xj is the position of the jth state variable, r and () is a random function generator between 0 and 1, Nc
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is the number of compensator capacitors and NF is a Step 8: Update change in location and position. To number of frogs. modify the position of each individual in the next stage, Step 3: Convert the constraint multi objective probit is necessary to evaluate the change in location of lem to an unconstraint one. each individual which is obtained from Equation (26). The proposed multi objective problem needs to be The position of each individual is modified by Equation converted into an unconstrained one by putting together (27). an augmented objective function and penalty factors: ⎛ ⎞ ⎛ ⎞⎤ ⎡ Neq Nueq ⎢ f1 (X) + L1 ⎝ (hj (X))2 ⎠ + L2 ⎝ (Max [0, −gj (X)])2 ⎠⎥ ⎢ ⎥ ⎢ ⎥ j=1 j=1 ⎢ ⎥ ⎡ ⎤ ⎢ ⎛ ⎞ ⎛ ⎞⎥ J1 (X) ⎢ ⎥ Neq Nueq ⎢ ⎢ ⎥ ⎥ 2⎠ 2 ⎠⎥ ⎢ ⎢ ⎥ ⎝ ⎝ (39) J(X) = ⎣ J2 (X)⎦ + L2 = ⎢ f2 (X) + L1 (hj (X)) (Max [0, −gj (X)]) ⎥ ⎢ ⎥ j=1 j=1 ⎥ J3 (X) 3×1 ⎢ ⎢ ⎛ ⎞ ⎛ ⎞⎥ ⎢ ⎥ Neq Nueq ⎢ ⎥ 2⎠ 2 ⎠⎦ ⎣ f3 (X) + L1 ⎝ ⎝ + L2 (hj (X)) (Max [0, −gj (X)]) j=1
Where, f1 (X), f2 (X) and f3 (X) are the objective function described in Equation (7) to (9). Neq and Nueq are the number of equality and inequality constraints, respectively. hj (X) and gj (X) are the equality and inequality constraints, respectively. L1 and L2 are the penalty factors. Since the constraints have to be met, the values of the parameters should be high. In this paper the values have been considered 10000. Step 4: Calculate the objective functions value and normalize them by fuzzy decision maker. The augmented objective function (Equation 39) is evaluated by using the result of the load flow. Also for each individual (Xi ) the membership values of all the different objectives are computed. Step 5: Apply the Pareto method to obtain normalized objective function of previous step and save non-dominate solutions in the repository. Compute the weight factor for all non-dominate solutions. Step 6: Divide sorted generated population in memeplexes according to their weight factors and select the best position in each memeplex (Xb ) and the best position among all memeplexes (Xg ). Step 7: A process is applied to improve only the frog with the worst fitness according to Equation (27), if the new generated individual dominates the worst solution then the worst one replaces by new generated individual otherwise a new individual is going to be generated in accordance with Equation (27) and compares with worst solution again, this process continues for a specific number of iterations (itetationmax1 ). If after itetationmax1 repetition there is no member that dominates the worst member afterward a new population is randomly generated to replace it.
j=1
Step 9: If any element of an individual breaks its inequality constraints then the position of the individual is fixed to its maximum/minimum operating point. Therefore, this can be formulated as:
k xi,j
⎧ k k x if xj,min xi,j xj,max ⎪ ⎪ ⎪ i,j ⎨ k = xj,min if xi,j xj,min ⎪ ⎪ ⎪ ⎩ k xj,max if xi,j xj,max
(40)
Where, i is the number of population and j ∈ {1, 2, . . . , Nparam } which Nparam is the number of control variables. Step 10: Compute the objective function value for the new generated solution, it operates as mentioned in step4. Step 11: Use mutation strategies. Select the ath strategy by the RWM based on the previous section for all the existing solutions. The new frog set must satisfy all the quality and inequality constraints. To handle these constraints, the steps 3 and 4 can be applied. Step 12: Apply the Pareto method to obtain normalized objective function and add the obtained non-dominate solutions to the repository. Step 13: Determine the non-dominate solutions between the repository members by using the Pareto method. Because of in the previous section some nondominate solutions were added to the repository and they might dominate solutions which had been existed in the repository thus it is necessary to apply the
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Pareto method to determine the non-dominate solutions between the new and old members of repository. Step 14: In this step all memeplexes combined and sort again. Step 15: If the current iteration number (iterationmax2 ) reaches the predetermined maximum iteration number, the search procedure is going to be stopped, otherwise goes to Step 6.
8. Simulation results The impact of the FACTS devices on fuel cost, loss and voltage stability index has been studied in this paper, to validate performance of the proposed method it has tested on the IEEE 30-bus test system [27]. The 30-bus system consists of six generators at buses #1, #2, #5, #8, #11 and #13. Also bus #1 is considered as the slack bus and voltage magnitude limits of all buses are set to 0.95 ≤ V ≤ 1.05. In this paper, XTCSC is limited between −0.2 and 0.2, and QSVC is restricted between 0 and 50. This paper includes four case studies to show the effectiveness of TCSC and SVC on each objective function individually and two or three objectives simultaneously. Comparisons of the proposed optimization algorithm with the corresponding other methods confirm the effectiveness of the proposed method. It is crucial to note that one TCSC is putted between buses #16 and #18 and one SVC is located on bus #10. Case 1. The best cost In this section just fuel cost is considered as objective function, for more validation the obtained result in case of FACTS absence is compared with other in literatures, also the cost in case of FACTS presence is shown, it is
clear that FACTS devices can decrease the generation cost in power system that help to increase in social welfare. Table 1 tabulates the best fuel cost obtained by proposed algorithm as compared to NLP [28], EP [29], TS [30], IEP [31], DE-OPF [32] and MDE-OPF [32]. It can be seen that the proposed approach performed better than above algorithms, this demonstrates potential and effectiveness of the proposed approach over the previous reported approaches. From Table 1 it is clear that total generation cost is decreased about 1.6 ($/hr) with FACTS devices, it is too important in state of economic problems, also it proves the FACTS reducing effect on fuel cost generation. For confirming the proposed method ability in finding the global optima, this algorithm is applied to find the best generation cost for 50 times, obtained results are compared with other algorithms. Figure 4 depicts the best and worst result, average and standard deviation for GA, PSO, SFLA and MSFLA for cost objective function. It is clear that MSFLA obtained a lower value and has a better standard deviation and worst result respect to other algorithms. Case 2. The voltage stability index Modern electrical equipment is designed to operate within a specific range of voltages. Equipment of both utility and customers, can only tolerate the fluctuations of voltage in a small range. If the voltage be far from its regular amount, the equipment has a high probability to get damaged, therefore voltage stability is very important parameter in power network. Our purpose is to determine the proper FACTS parameter to maximize the system voltage improvement, which is evaluated by voltage stability index. Table 2 depicts the obtained results and compare it with other methods in literatures such as EGA-DQLF [33], PSO [34] and FAPSO [34]
Table 1 Result of cost objective function for different method with and without FACTS devices PG1 (MW) PG2 (MW) PG5 (MW) PG8 (MW) PG11 (MW) PG13 (MW) Total generation (MW) TCSC parameter SVC parameter Cost ($/h)
NLP [28]
EP [29]
TS [30]
IEP [31]
DE-OPF [32]
MDE-OPF [32]
MSFLA
MSFLA with FACTS
176.26 48.84 21.51 22.15 12.14 12 292.9 – – 802.4
173.848 49.998 21.386 22.63 12.928 12 292.79 – – 802.62
176.04 48.76 21.56 22.05 12.44 12 292.85 – – 802.29
176.2358 49.0093 21.5023 21.8115 12.3387 12.0129 292.9105 – – 802.465
176.009 48.801 21.334 22.262 12.46 12 292.866 – – 802.394
175.974 48.884 21.51 22.24 12.251 12 292.859 – – 802.376
179.1929 48.9804 20.4517 20.9264 11.5897 11.9579 293.0991 – – 802.287
178.315 47.8558 21.313 21.217 11.8270 12 292.5287 −0.195 41.235 800.6858
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Fig. 4. Comparison of average and standard deviation for 50 trails (Cost). Table 2 Result of voltage stability index objective function for different method with and without FACTS devices VG (1) VG (2) VG (5) VG (8) VG (11) VG (13) T6-9 (p.u) T6-10 (p.u) T4-12 (p.u) T27-28 (p.u) TCSC parameter SVC parameter Voltage stability index
Initial [33]
EGA-DQLF [33]
PSO [34]
FAPSO [34]
MSFLA
MSFLA with FACTS
1 1 1 1 1 1 1 1 1 1 – – 0.23
1.0618 1.053 1.053 1.014 1.025 1.046 0.9125 0.9 0.9 0.925 – – 0.10402
– – – – – – – – – – – – 0.1307
– – – – – – – – – – – – 0.1238
1.0498 1.0495 1.0479 1.0498 1.0431 1.0497 0.95 0.92 0.91 0.90 – – 0.1037
1.05 1.05 1.0493 1.05 1.0489 1.0495 0.94 0.92 0.91 0.91 −0.1107 23.56 0.1004
from this table it is clear that FACTS devices effectively improved the voltage stability index. From comparing the objective in absence and presence of FACTS devices, it can be seen that the voltage stability index is improved from 0.1037 to 0.1004. For the given 30-bus system, the voltage stability index is about 0.1037 for the MSFLA algorithm without FACTS devices whereas, in SFLA the result is 0.1043 for the same population and iteration size. It proves the mutation ability to find the global minima that reinforce the MSFLA search ability.
Case 3. The best loss Transmission loss minimization is considered as objective function in this section, for depicting the effect of TCSC and SVC on transmission loss, obtained results have compared with SPEA and NSGA II [35] without FACTS devices that are shown in Table 3. With comparing the objectives in absence and presence of the FACTS devices, it can be seen that the total loss decreases from 5.2732 Mw to 4.952 MW.
Case 4. Minimize objective functions simultaneously In this case fuel cost, voltage stability index and loss are considered two by two simultaneously. These multi-objective optimization problems are solved by the proposed approach in cases of FACTS presence and absence. The Pareto optimal solution obtained using the proposed algorithm with and without FACTS devices are depicted in Figs. 5 to 7. From the Pareto optimal solution, it is clear that the proposed MSFLA method is able to give well distributed solutions. Pareto fronts for case of FACTS presence are shown with blue color. The obtained solution in case of FACTS presence are dominated the solution of FACTS absence and it convey the idea that FACTS devices can improved the power system characteristic. In this paper, the optimal parameter setting of TCSC and SVC devices are optimized to minimize the fuel cost, loss and voltage stability index individually and simultaneously. The parameter of TCSC and SVC devices are very impressive on their applications, therefore in this paper they are optimized properly subject
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R. Azizipanah-Abarghooee et al. / Modified shuffled frog leaping algorithm for multi-objective optimal power flow Table 3 Result of loss objective function for different method with and without FACTS devices
VG (2) VG (5) VG (8) VG (11) VG (13) T6-9 (p.u) T6-10 (p.u) T4-12 (p.u) T27-28 (p.u) TCSC parameter SVC parameter Power loss (MW)
Initial [35]
SPEA [35]
NSGA II [35]
MSFLA without FACTS
MSFLA with FACTS
1.045 1.01 1.01 1.05 1.05 0.97 0.96 0.93 0.96 – – 5.4356
1.044 1.023 1.022 1.042 1.043 1.09 0.90 1.02 0.96 – – 5.1995
1.03 1.00 1.00 1.02 1.04 1.00 1.01 1.00 1.04 – – 5.3513
1.041 0.976 1.035 0.984 1.042 0.99 0.98 1.01 1.03 – – 5.2732
1.046 1.039 1.042 1.013 0.971 1.02 1.05 1.08 0.97 −0.186 25.73 4.952
Fig. 5. Pareto front for cost and loss objectives with and without FACTS devices.
to minimize each objectives individually and simultaneously. Finally, our results show that using FACTS devices with optimal parameter settings can significantly improve the characteristic of power systems. All three objectives with and without FACTS devices are optimized simultaneously, obtained results are shown in Figs. 8 and 9. From comparing these figures it is clear that obtained results for FACTS equipped power system are close to their single optimization value, also in both figures Pareto fronts are well distributed. Figures 8 and 9 once again prove that the proposed method is giving well distributed Pareto optimal front and achieve values close to single optimization values, also the best compromise solution is determined in each figure. The advantages of the projected model have been demonstrated in the entire previous sections and accomplished in Section 9. Beside, one of the main disadvantages of the suggested MSFLA technique in comparison with the classical optimization method such as linear programming [11] and etc is its high requirement for population and iteration process.
Fig. 6. Pareto front for cost and voltage stability index objectives with and without FACTS devices.
Fig. 7. Pareto front for loss and voltage stability index objectives with and without FACTS devices.
They increase with the number of the FACTS devices which are used in the transmission networks. However, it is noteworthy that these methods lead to non-optimal solution. It should be pointed out that another disadvantage of the meta-heuristic evolutionary algorithms is their less robustness in achieving optimal solution. The simulation results demonstrate that the robustness of the proposed MSFLA is acceptable.
R. Azizipanah-Abarghooee et al. / Modified shuffled frog leaping algorithm for multi-objective optimal power flow
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Fig. 8. Pareto front for cost, loss and voltage stability objectives with FACTS devices.
Fig. 9. Pareto front for cost, loss and voltage stability objectives without FACTS devices.
9. Conclusion This paper evaluates the effectiveness TCSC and SVC devices in decreasing the generation cost and total loss and improving the voltage stability index in power systems. Also this paper presents a new method to optimize objective functions individually and simultaneously that called MSFLA. This multi-objective optimization problem obtained all non-dominate Pareto optimal solution, to find optimal parameters of SVC and TCSC. Finally, a fuzzy membership approach has been used to identify the best compromise solution. From the simulation results, it is clear that proposed MSFLA method is able to give well distributed Pareto optimal solutions for proposed problem. It is clear that all Pareto solutions
when TCSC and SVC are in the power system dominated solutions which FACTS devices are not in power system, it proves the FACTS devices effect on cost, loss and voltage stability index. The strength of this method is not imposing any limitation on the number of objectives and achieves a set of solution instead one solution.
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