Multi-objective GA with fuzzy decision making for ... - Semantic Scholar
Applied Soft Computing 12 (2012) 2756–2764
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Multi-objective GA with fuzzy decision making for security enhancement in power system R. Narmatha Banu ∗ , D. Devaraj Department of Electrical Engineering, Kalasalingam University, Krishnankoil, Tamil Nadu, India
a r t i c l e
i n f o
Article history: Received 18 January 2011 Received in revised form 10 August 2011 Accepted 7 March 2012 Available online 30 April 2012 Keywords: Power system security Flexible AC transmission system (FACTS) devices Thyristor Controlled Series Capacitors (TCSC) Genetic algorithm Pareto optimal frontier
a b s t r a c t Power system security enhancement is a major concern in the operation of power system. In this paper, the task of security enhancement is formulated as a multi-objective optimization problem with minimization of fuel cost and minimization of FACTS device investment cost as objectives. Generator active power, generator bus voltage magnitude and the reactance of Thyristor Controlled Series Capacitors (TCSC) are taken as the decision variables. The probable locations of TCSC are pre-selected based on the values of Line Overload Sensitivity Index (LOSI) calculated for each branch in the system. Multi-objective genetic algorithm (MOGA) is applied to solve this security optimization problem. In the proposed GA, the decision variables are represented as floating point numbers in the GA population. The MOGA emphasize non-dominated solutions and simultaneously maintains diversity in the non-dominated solutions. A fuzzy set theory-based approach is employed to obtain the best compromise solution over the trade-off curve. The proposed approach has been evaluated on the IEEE 30-bus and IEEE 118-bus test systems. Simulation results show the effectiveness of the proposed approach for solving the multi-objective security enhancement problem. © 2012 Elsevier B.V. All rights reserved.
1. Introduction In any power system, unexpected outage of transmission lines occurs due to faults or other disturbances. These events referred to as contingencies, may cause significant overloading of transmission lines, which in turn may lead to total or partial system blackout. Security constrained optimal power flow (SCOPF) is the main tool used in the energy control centers to avoid limit violation in the contingency state. SCOPF [1] adjusts base case decision variables to minimize the defined objective function subject to base case and contingency state operating constraints. The solution of an SCOPF is useful for both system operation and planning. The SCOPF does not take advantage of the post-contingency corrective rescheduling that is possible in static security enhancement. In [2], a mathematical framework was proposed for the solution of the SCOPF problem taking into account the system corrective capabilities such as generation rescheduling after the outage has occurred. The resulting dispatch has the same security level as the SCOPF, but with lower operating costs. An iterative approach is proposed in [6] to solve the SCOPF with corrective action.
∗ Corresponding author at: Kalasalingam University, Department of Electrical Engineering, Anandnagar, Krishnankoil 626126, Virudhunagar, Tamil Nadu, India. Tel.: +91 4563 289042; fax: +91 4563 289322. E-mail addresses: [email protected] (R.N. Banu), [email protected] (D. Devaraj). 1568-4946/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.asoc.2012.03.057
Apart from generation re-scheduling, FACTS devices [3,4] based on power electronics technology can also be used for power flow control through transmission lines. Thyristor Controlled Series Capacitors (TCSC), one of the FACTS devices can be used effectively in alleviating the line overload in case of a contingency. In this work, the base case generator active power, generator bus voltages and contingency state TCSC reactance values are used as the decision variables for security enhancement. For a large-scale power system, more than one FACTS device may have to be installed to achieve the desired performance. Studies have been conducted to identify the suitable location for FACTS devices to improve power system security. In this work, the location of TCSC is identified based on line overload severity index computed for every line in the system. While using FACTS devices for the performance improvement of power system, the installation cost need to be taken into account which is not done in the above papers. In this work, the installation cost of TCSC is taken as the additional objective of the OPF problem. In the literature, the SCOPF with corrective action is treated as a single-objective optimization problem [5,6]. In this paper, the SCOPF with corrective action is treated as a true multi-objective optimization problem with minimization of fuel cost and the installation cost of TCSC as the objectives. Because of the presence of conflicting multiple objectives, a multi-objective optimization problem results in a number of optimal solutions, known as pareto optimal solutions [7,8]. In a multi-objective optimization, effort must be made in finding the set of trade-off optimal solutions by considering all objectives to be important.
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Gij , Bij Gii , Bii Gk FT NB NB−1 NPQ Ng Nl Pi , Qi Pgi , Qgi Sl Slmax Vi Vj ij
mutual conductance and susceptance between bus i and bus j self-conductance and susceptance of bus i conductance of branch k total fuel cost total number of buses total number of buses excluding slack bus number of PQ buses number of generator buses number of branches in the system real and reactive powers injected into network at bus i real and reactive power generation at bus i apparent power flow through the lth branch apparent power flow limit through the lth branch voltage magnitude at bus i voltage magnitude at bus j voltage angle difference between bus i and bus j
One, straightforward approach to solve the multi-objective optimization problem is to convert them into a single objective problem by linear combination of different objectives as a weighted sum and then solve it similar to single objective optimization problems [9]. The important aspect of this weighted sum method is that a set of non-inferior (or pareto optimal) solutions can be obtained by varying the weights. Unfortunately, this requires multiple run as many times as the number of desired pareto optimal solutions. Furthermore, this method cannot be used to find pareto optimal solutions in problems having a non-convex pareto optimal front. To avoid this difficulty, the Є-constraint method [10] is used for multi-objective optimization problem. This method is based on optimizing the most preferred objective and considering the other objectives as constraints bounded by some allowable levels. These levels are then altered to generate the entire pareto optimal set. This approach is time-consuming and tends to find weak pareto optimal solutions. The ability of Evolutionary Computation techniques like Genetic Algorithm to find multiple optimal solutions in one single simulation run makes them unique in solving multi-objective optimization problems [7]. In this work, the multi-objective security optimization problem is solved using multi-objective genetic algorithm (MOGA) [11]. Like the other approaches such as NSGA II, SPEA2, IBEA and DEMO, MOGA is also a population-based search algorithm. Each algorithm differs in the way fitness value is assigned to the individuals while solving the multi-objective optimization problem. The environmental selection in NSGA-II [12] first ranks the individuals using non-dominated sorting. To distinguish between individuals with the same rank, the crowding distance metric is used, which prefers individuals from less crowded regions of the objective space. SPEA2 [13] works similarly, calculating the raw fitness of the individuals according to Pareto dominance relations between them and using a density measure to break the ties. The individuals that reside close together in the objective space are discouraged from entering the archive of best solutions. IBEA [14], on the other hand, uses a different approach. The fitness of individuals is determined only according to the value of a predefined indicator. This indicator has to be dominance preserving and no other explicit diversity preserving mechanism (such as crowding in NSGA-II or density in SPEA2) is applied. In DEMO (Differential Evolution for Multi-objective Optimization) [15], the fitness of an individual is first calculated using Pareto-based ranking and then reduced with respect to the individual’s crowding distance value. This single
-jxc
i
Nomenclature
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rij+jxij
j
Fig. 1. Equivalent circuit of TCSC.
fitness value is then used to select the best individuals for the new population. Generally, binary strings are used to represent the decision variables of the optimization problem in the genetic population irrespective of the nature of the decision variables. This binarycoded GA has Hamming cliff problems [17] which sometimes may cause difficulties in the case of coding continuous variables. Also, for discrete variables with total number of permissible choices not equal to 2k (where k is an integer) it becomes difficult to use a fixed length binary coding to represent all permissible values. To overcome these difficulties, in this paper, the optimization variables namely, generator active power generation Pgi , generator bus voltages Vgi and TCSC settings are represented as floating point numbers in the genetic population. For effective genetic operation, crossover and mutation operators which can directly operate on floating point numbers [24] are used. The effectiveness and potential of the proposed approach to solve the multi-objective optimal power flow (OPF) problem has been demonstrated using IEEE 30bus and IEEE 118-bus systems. Lesser computational time taken by the MOGA to reach the optimal solutions makes it suitable for solving the large scale optimization problem like SCOPF. 2. Modelling and placement of Thyristor Controlled Series Capacitors (TCSC) Thyristor Controlled Series Capacitors (TCSC) consist of a fixed capacitor in parallel with a thyristor controlled reactor. The primary function of the TCSC is to provide variable series compensation to a transmission line. This changes the line flow due to change in series reactance. Fig. 1 shows a model of transmission line with TCSC connected between buses ‘i’ and ‘j’. For steady state analysis, the TCSC can be considered as a static reactance −jxc . The controllable reactance xc is directly used as the control variable in the power flow equations. The power flow equations of a transmission line with TCSC can be written as: Pij = Vi2 gij − Vi Vj (gij cos ıij + bij sin ıij ) Qij = −Vi2 bij − Vi Vj (gij sin ıij − bij cos ıij )
(1)
where rij
gij =
rij2 + (xij − xc )2 xc bij = xij − rij2 + (xij − xc )2
(2)
The only difference between normal line power flow equation and the TCSC line power flow equation is the presence of the controllable reactance xc which is varied by adjusting the value of TCSC reactance. To enhance the security of the system, the TCSC has to be placed at the suitable locations. To determine the best location of TCSC, an index called Line Overload Sensitivity Index (LOSI) is calculated for all the remaining lines. The LOSIl for branch “l” is defined as the sum of the normalized power flow through branch “l” to all the considered contingencies ‘C’, expressed as: LOSIl =
NC SlC C=1
Slmax
(3)
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where SlC = MVA flow in line ‘l’ during contingency “C”. The LOSI defined at branch “l” for the base case loading is defined by LOSIlBL . In order to achieve optimal location of TCSC, valid under change in system loading, LOSI indices defined in (4) are also computed at an increased loading and decreased loading scenario. The increased loading scenario pertains to all the loads increased by 5% from their base values and the decreased loading scenario has been simulated with the loads decreased by 5% from their base values. The corresponding LOSI, calculated at each overloaded lines, are termed as LOSIlIL and LOSIlDL respectively. The optimal location of TCSC has been decided by an average line overload severity index, computed for every line, as defined in the following:
LOSIl =
LOSIlBL + LOSIlIL + LOSIlDL 3
(4)
The branches are ranked based on their corresponding LOSIl values. The TCSC are placed on the branches starting from the top of the ranking list and proceeding downward with as many branches as the number of available TCSC. 3. Problem formulation
3.1. Objective functions 3.1.1. Economic objective function (FC ) The most commonly used objective in the OPF problem formulation is the minimization of the total operation cost of the fuel consumed for producing electric power within a scheduled time interval. The individual costs of each generating unit are assumed to be function only of real power generation and are represented by quadratic curves. The objective function for the entire power system can be expressed as the sum of the quadratic cost model at each generator [18]. Ng
2 (ai Pgi + bi Pgi + ci ) ($/h)
(5)
i=1
where ai , bi , and ci are the cost coefficients of generator at bus i. Pgi is the active power generation at bus i. 3.1.2. TCSC cost function (FE ) It is important to take the economical aspects of the TCSC devices installed in the power system due to high investment and operating costs. The total investment cost of the TCSC device is expressed as [21]: FIC =
(fi + vi Xi )
• load flow constraints Qi − Vi
NB
Vj (Gij sin ij − Bij cos ij ) = 0,
(6)
i∈˝
where fi and vi are the fixed cost and variable cost for candidate. ˝ is a set of all candidate sites, and Xi , is the rating of TCSC device i.
i = 1, 2, . . . , NPQ
j=1
(7)
Pi − Vi
NB
Vj (Gij cos ij + Bij sin ij ) = 0,
i = 1, 2, . . . , NB−1
(8)
j=1
• voltage constraint Vimin ≤ Vi ≤ Vimax ,
i ∈ NB
(9)
• real power generation limit Pgmin ≤ Pg ≤ Pgmax ,
g ∈ Ng
(10)
• generator reactive power generation limit Qgmin ≤ Qg ≤ Qgmax ,
In general, the optimal power flow (OPF) problem is formulated as an optimization problem in which one or more objective functions are minimized while satisfying a number of equality and inequality constraints. In the security enhancement problem considered here the goal is to determine the optimal values of generator active power, generator bus voltage magnitudes and TCSC that enhance the systems security level while minimizing the generator fuel cost and investment cost of TCSC. Minimization of fuel cost will necessitate higher values of TCSC to reach the same level of security. This will be lead to increased value of TCSC installation cost. The mathematical formulation of the security enhancement problem is given below.
FC =
3.1.3. Problem constraints The constraints are:
g ∈ Ng
(11)
• transmission line flow limit Sl < Slmax ,
l ∈ Nl
(12)
• limit on reactance of TCSC min max ≤ XTCSC,i ≤ XTCSC,i , XTCSC,i
i ∈ NTCSC
(13)
The security constrained optimal power flow (SCOPF) formulation considers both the pre and post contingency state power flows and all constraints in those states should always be satisfied. The post-contingency constraints are of the same dimension as those of the pre-contingency case. If there are m total constraints in a given base case optimal power flow (OPF), there will be (k + 1)m constraints in a SCOPF formulation with k contingencies. Aggregating the objectives and constraints, the problem can be mathematically formulated as a non-linear constrained multiobjective optimization problem as follows: minimize FT = [FC , FIC ]
(14)
subject to the constraints (7)–(13). 4. Multi-objective genetic algorithm Genetic algorithms (GA) [22] are generalized search algorithms based on the mechanics of natural genetics. GA maintains a population of individuals that represent the candidate solutions to the given problem. Each individual in the population is evaluated to give some measure of its fitness to the problem from the objective function. GAs combine solution evaluation with stochastic genetic operators namely, selection, crossover and mutation to obtain near optimality. Being a population-based approach, GA is well suited to solve multi-objective optimization problems. Multi-objective genetic algorithm [7,23] is an extension of classical GA. The main difference between a conventional GA and a MOGA lies in the assignment of fitness to an individual. The rest of the algorithm is the same as that in a classical GA. The details of the MOGA are described below. In the MOGA, first, each solution is checked for its domination in the population. Two solutions (x(1) and x(2) ) are compared on the basis of whether one dominates the other solution or not. A solution x(1) is said to dominate the other solution x(2) , if the following conditions are satisfied:
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(a) The solution x(1) is no worse than x(2) in all objectives, or ¯ i (x(2) ) for all i = 1, 2, . . ., M where M be the objective fi (x(1) )f functions. (b) The solutions x(1) is strictly better than x(2) in at least one objective, or fi (x(1) ) fi (x(2) ) for at least one j (j ∈ {1, 2, . . ., M}) If any of the above condition is violated, the solution x(1) does not dominate the solution x(2) (or mathematically x(1) ≤ x(2) ). To a solution ‘i’, a rank ri equal to one plus the number of solutions i that dominate solution ‘i’ is assigned: ri = 1 + i
(15)
In this way, non-dominated solutions are assigned a rank equal to 1, since no solution would dominate a non-dominated solution in a population. Once the ranking is done, a raw fitness is assigned to each solution based on its rank. To perform this, first the ranks are sorted in ascending order of magnitude. Then, a raw fitness is assigned to each solution by using a mapping function. Usually, the mapping function is chosen so as to assign fitness between N (for the best rank solution) and 1 (for the worst rank solution). Thereafter, solutions of each rank are considered at a time and their raw fitness are averaged. This average fitness is called the assigned fitness to each solution of the rank. This process emphasizes non-dominated solutions in the population. In order to maintain diversity among non-dominated solutions, niching among solutions of each rank are introduced. The niche count is calculated with the following equation: nci =
(ri )
Sh(dij )
(16)
j=1
where (ri ) is the number of solutions in rank ri and Sh(dij ) is the sharing function of two solutions i and j. The sharing function Sh(d) is calculated using objective function value as distance metric as Sh(dij ) =
⎧ ⎨ ⎩
1−
˛
dij share
,
0,
if d ≤ share
k=1
j
fki − fk fkmax − fkmin
2 (18)
where fkmax and fkmin are the maximum and minimum objective function value of the kth objective. The shared function takes a value in [0, 1], depending on the values of dij and share . The shared fitness value is calculated by dividing the assigned fitness of a solution by its niche count. Although all solutions of any particular rank have the identical fitness, the shared fitness value of a solution residing in a less crowded region has a better shared fitness. This produces a large selection pressure for poorly represented solutions in any rank. Dividing the assigned fitness value by the niche count reduces the fitness of each solution. In order to keep the average fitness of the solutions in a rank the same as that before sharing, these fitness values are scaled using Eq. (19) so that their average shared fitness value is the same as the average assigned fitness value. fjSC =
fj (r)
(r)
f k=1 k
culated using fj = fj /ncj ; (r) is the number of solutions in rank ri . This procedure is continued until all ranks are processed. Thereafter, selection, crossover and mutation operators are applied to create a new population. With each individual represented as a string of integers and floating point numbers, selection process remains the same as in classical GA, but the cross over and mutation operators are applied variable by variable. In this paper, tournament selection and BLX-␣ crossover and non-uniform mutation operators [7] are used.
5. Best compromise solution Upon having the pareto optimal set of non-dominated solution, it is preferred to get the best compromise solution for implementation. The Many-objective Distinct Candidates Optimization using Differential Evolution (MODCODE) algorithm [16] discovered a low number of solutions within a region of interest on the true pareto front. It aims at returning a few optimal distinct solutions within a region of interest, with both result set cardinality and distinctiveness being user defined in compliance with the goals of MODCO algorithm. Considering the imprecise nature of decision maker’s judgment this work applies a fuzzy set-based approach to obtain the best compromise solution [19]. Fuzzy set theory generalizes classical set theory to allow partial membership with a smooth boundary. The degree of membership in a set is expressed by a number between 0 and 1.0 means entirely not in the set, 1 means completely in the set, and a number in between 0 and 1 means partially in the set. In this paper, it is assumed that the decision maker (DM) has fuzzy goal for each objective. These fuzzy goals depend on experience and intuitive knowledge of the DM. The fuzzy goals are represented by linear membership function as given by Eq. (20). The ith objective function Fi is represented by a membership function i defined by
i (Fi ) =
where ‘ share ’ is the sharing parameter which signifies the maximum distance between any two solutions before they can be considered to be in the same niche and dij is the normalized distance between any two solutions i and j in a rank. The normalized distance dij is calculated using
M dij =
where fjSC is the scaled fitness; fj is the shared fitness and it is cal-
(17)
otherwise
(19)
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⎧ 1, ⎪ ⎨ F max − F
i
i
max min ⎪ ⎩ Fi − Fi
0,
Fi ≤ Fimin , Fimin < Fi < Fimax Fi ≥
(20)
Fimax
where Fimin and Fimax are the minimum and maximum value of the ith objective function respectively among all non-dominated solutions. The value of membership function suggests how far (in the scale from 0 to 1) a non-inferior (non-dominated) solution has satisfied the Fi objective. The sum of membership function values (Fi ) (i = 1, 2, 3, . . ., M) for all the objectives can be computed in order to measure the accomplishment of each solution in satisfying the objectives. The accomplishment of each non-dominated solutions can be rated with respect to all the N non-dominated solutions by normalizing its accomplishment over the sum of the accomplishments of N non-dominated solutions as follows:
M
k
=
i=1
(Fik )
N M k=1
i=1
(Fik )
(21)
The solution that attains the maximum membership k , in the fuzzy set so obtained can be chosen as the best solution or the one having the highest cardinal priority ranking. 6. Genetic algorithm implementation While applying GA for solving the SCOPF problem, the following issues need to be addressed:
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Table 1 Line outage ranking using severity index. Outage line no.
Over loaded lines
Line flow (MVA)
Line flow limit (MVA)
Severity index (SI)
Rank
1–2
1–3 3–4 4–6
191.58 174.13 103.37
130 130 90
5.262
1
1–3
1–2 2–6
181.17 66.482
130 65
3.010
2
3–4
1–2 2–6
178.43 65.558
130 65
2.9011
3
22–24 24–25
19.062 17.781
16 16
2.1979
4
2–6 5–7
76.285 101.08
65 70
1.3777
5
28–27 2–5
• solution representation and • fitness evaluation.
⎧ ⎨ KV (ViK − Vimax )2 , if Vik > Vimax
PViK =
2
k K min min ⎩ KV (Vi − Vi ) , if Vi < Vi
0,
6.1. Solution representation Implementation of GA for a problem starts with the parameter encoding (i.e., the representation of the problem). Each individual in the genetic population represents a candidate solution. The elements of that solution consist of all the decision variables in the system. The decision variables of the SCOPF problem include active power generation Pgi , generator bus voltage magnitude Vgi and transformer tap settings tk . The solution variables are represented as floating point numbers and integers. This representation has a number of advantages over binary coding. The efficiency of the GA is increased as there is no need to convert the solution variables to the binary type. Also, the computer memory required to store the population is reduced.
PQiK =
otherwise
⎧ ⎨ Kq (QiK − Qimax )2 , if Qik > Qimax 2
k K min min ⎩ Kq (Qi − Qi ) , if Qi < Qi
PLlK =
(24)
0,
(25)
otherwise
2
Kl (SlK − Slmax ) , 0,
if Slk > Slmax otherwise
(26)
where Ks , Kv , Kq , and Kl are the penalty factors for slack bus power output, bus voltage limit violation, generator reactive power limit violation and line flow violation respectively. The penalty function is added to each of the objective function to get the new objective functions. 7. Simulation results
6.2. Evaluation function GA searches for the optimal solution by maximizing a given fitness function, and therefore an evaluation function which provides a measure of the quality of the problem solution must be provided. In the SCOPF problem under consideration, the objectives are to minimize the fuel cost of generation and investment cost of TCSC satisfying the constraints. The equality constraints given by Eqs. (7) and (8) are satisfied by running the power flow program. The active power generation (Pgi ) (except the generator at the slack bus), generator terminal bus voltages (Vgi ), transformer tap settings (tk ) and reactance of TCSC (XTCSC ) are the decision variables and they are self-restricted by the optimization algorithm. The limit on active power generation at the slack bus (Pgs ), load bus voltages (Vload ), reactive power generation (Qgi ) and line flow (Sl ) are satisfied by adding a penalty function with the objective function. With the inclusion of the penalty terms, the over all penalty function becomes: PF =
NC
PS K +
K=0
NC NPQ
PViK +
K=0 i=1
NC Ng
PQiK +
K=0 i=1
NC Nl
PLlK
(22)
K=0 l=1
where PSK , PViK , PQiK and PLlK are the penalty terms for the slack bus generator active power limit violation, load bus voltage-limit violation, reactive power generation limit violation, and line flow limit violation respectively. These quantities are defined by the equations:
PS K =
⎧ ⎨ KS (PSK − PSmax )2 , if PSK > PSmax 2
K min K min ⎩ KS (PS − PS ) , if PS < PS
0,
otherwise
(23)
The proposed multi-objective genetic algorithm approach has been applied to solve the security enhancement problem in IEEE30 bus and IEEE 118-bus test systems. The IEEE 30-bus system has 6 generator buses, 24 load buses and 41 transmission lines, of which 4 branches (6–9), (6–10), (4–12) and (28–27) are with tap setting transformers. The generator and transmission-line data relevant to the system are taken from [1]. The upper and lower voltage limits at all the bus bars except slack bus are taken as 1.10 p.u. and 0.95 p.u. respectively. The slack bus bar voltage is fixed to its specified value of 1.06 p.u. The generator cost coefficients and the transmission line parameters are taken from [1]. The IEEE 118-bus test system has 54 generator buses and 186 transmission lines. All other data are the same as the standard IEEE 118-bus data given in [20]. To demonstrate the effectiveness of the proposed approach, three different cases have been considered as follows: Case 1: SCOPF with minimization of fuel cost as objective in IEEE 30-bus system without considering post contingency rescheduling. Case 2: Multi objective optimal power flow for security enhancement in IEEE 30-bus system. Case 3: Multi objective optimal power flow for security enhancement in IEEE 118-bus system. 7.1. Case 1: SCOPF with minimization of fuel cost as objective in IEEE 30-bus system In this case, contingency analysis was carried out on the system to identify the severe contingencies. The list of severe contingencies along with their severity index value which were from Eq.
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Table 4 Extreme solutions for the three severe contingencies. Control variables
Minimum fuel cost solution
Minimum installation cost solution
P1 P2 P5 P8 P11 P13 V1 V2 V5 V8 V11 V13 TCSC (line outage 1–2)
124.92 58.64 25.32 34.89 22.934 20.67 1.0489 1.0345 1.0096 1.0147 1.0986 1.0580 −0.5000, −0.0734 −0.5000, −0.2441 −0.4173, −0.4336 814.7269 3.56 × 104
129.74 44.121 40.507 27.471 25.984 17.508 1.0476 1.0322 1.002 1.0023 1.0597 1.0566 −0.5000, −0.4958 −0.3468, −0.4709 −0.2895, −0.2612 827.5058 0.47 × 104
TCSC (line outage 1–3) TCSC3 (line outage 3–4)
Fig. 2. Convergence of the GA-SCOPF algorithm for IEEE 30-bus test system.
(A.1) is given in Table 1. From the contingency analysis, it is found that five contingencies have resulted in overload on other lines. The line flows corresponding to the severe contingencies were included as additional constraints of the OPF problem and the proposed GA was applied to solve this security constrained OPF problem with the minimization of base case fuel cost as the objective function. Generator active power outputs and the generator bus terminal voltages were taken as the optimization variables. The optimization variables are represented as floating point numbers in the GA population. The initial population was randomly generated between the variable’s lower and upper limits. Tournament selection was applied to select the members of the new population. Blend crossover and uniform mutation were applied on the selected individuals. The performance of GA generally depends on the GA parameters used, in particular, the crossover and mutation probabilities, Pc and Pm, respectively. The performance of GA for various crossover and mutation probabilities in the range of 0.6–0.9 and 0.001–0.01 respectively was therefore evaluated. The best result of the GA was obtained with the following algorithm parameters:
Fuel cost ($/h) Installation cost ($)
Table 5 Best compromise solution for the IEEE 30-bus system. Control variables
Optimal Control variable settings
P1 , P2 , P5 , P8 , P11 , P13 (base)
125.265,58.645, 25.317, 34.89, 22.93, 20.67 0.9750, 1.0500, 1.1000, 0.9800, 0.9670, 0.9520, 1.0140, 1.0750, 0.9840. −0.5000, −0.0404 −0.4900, −0.321 −0.323, −0.4321 819.41 1.52 × 104
V1 , V2 , V5 , V8 , V11 , V13 TCSC (line outage 1–2) TCSC (line outage 1–3) TCSC3 (line outage 3–4) Fcost ($/h) Installation cost ($)
No. of generations: 100, population size: 50, crossover probability: 0.9, mutation probability: 0.01. After 100 generations it was found that all the individuals have reached almost the same fitness value. This shows that GA has reached the optimal solution. Fig. 2 shows the variation of fitness during the GA run for the best case. Ten trail runs of the GA was conducted for the SCOPF problem and the minimum, maximum and the average value of fuel cost are 824.8991 $/h, 824.991 $/h and 824.9231 $/h respectively.
Table 2 Result of SCOPF Algorithm. P1 P2 P5 P8 P11 P13 V1 V2 V5 V8 V11 V13
142.38 49.18 29.51 34.59 16.70 20.6 1.0498 1.0340 1.0105 1.0245 0.9310 1.0264
Generation cost
824.9868 ($/h)
Table 3 LOSI values for IEEE 30 bus test system. S. no.
Branches
LOSI value
1 2
2–6 2–4
0.93942 0.3283
Fig. 3. Pareto-optimal front for the three severe contingencies.
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Table 6 Line outage ranking using severity index for IEEE 118-bus system. Outage line no.
Over loaded lines
Line flow (MVA)
Line flow limit (MVA)
Severity index (SI)
Rank
8–5
30–17 8–30
506.554 419.466
500 200
5.4252
1
9–10
8–30 30–38
214.225 198.765
200 175
3.1158
2
8–9
8–30 30–38
270.172 198.713
200 175
3.1142
3
The minimum cost obtained by the GA based approach along with the optimal decision variables are given in Table 2. The algorithm took 129 s to reach the optimal solution. Corresponding to these decision variables, it was found that all the state variables satisfy the lower and upper limits. The secured optimal solution obtained by this algorithm does not violate any constraints, whereas the optimal solution reported in [1] violates line loading limit under contingencies 1–2 and 28–27 and the results given in [25] violates some line loading limit under contingency 1–2. Further, the fuel cost obtained by the proposed algorithm is less than the value reported in [26] for the same IEEE 30-bus system. 7.2. Case 2: multi-objective SCOPF in IEEE 30-bus system In this case, the multi-objective genetic algorithm was applied to enhance the security of the system for the first three contingencies. The LOSI values are calculated using (3) for each branch of the studied system for the severe contingencies. The top two branches which posses high values of LOSI are listed in Table 3. The TCSC are placed in these two lines. It is assumed that the impedance of all TCSC can be varied within 50% of the corresponding branch impedance. Generator active power, generator bus bar voltages and the reactance values of TCSC are taken as the decision variables and the problem was handled as a multi-objective optimization problem with TCSC installation cost and generator fuel cost as objectives to be minimized simultaneously. For the considered power system, the MOGA was applied considering several sets of parameters in order to prove its capability to provide acceptable trade-offs close to the Pareto Optimal Front (POF). In all simulation, the following parameters were used: Number of generation: 100, population size: 50, cross over probability: 0.9 and mutation probability: 0.01. The pareto optimal front obtained for the three contingency cases are plotted in Fig. 3. It is worth mentioning that the proposed approach has produced 16 pareto optimal solutions in a single run
Table 7 LOSI values for IEEE 118-bus test system. S. no.
Branches
LOSI value
1 2 3
8–30 30–38 30–17
1.2208 0.8358 0.7199
that have satisfactory diversity characteristics and span the entire pareto optimal front. Out of these, two optimal solutions which are the extreme points of Fig. 3 that represent the best installation cost and best fuel cost are given in Table 4. The best compromise solutions which were found from Eqs. (20) and (21) are given in Table 5. In all cases, the value of SI is zero, which shows that the proposed approach is able to alleviate the line overload.
Fig. 4. Pareto-optimal front for the severe contingency.
Table 8 Pareto optimal solutions for the most severe contingencies. Control variables
Minimum fuel cost solution
Minimum installation cost solution
P10 , P12 , P25 , P26 , P31 , P46 , P49 , P54 , P59 , P61 , P65 , P66 , P69 , P80 , P87 , P89 , P100 , P103 , P111
201.8711, 108.6736, 219.7154, 264.0657, 19.7902, 28.5594, 213.0799, 63.3050, 199.4073, 156.4032, 331.5864, 231.4445, 201.6571, 405.5204, 12.7802, 215.6403, 223.2229, 35.7076, 52.6144
220.8711, 118.76, 148.7154, 283.0657, 53.87, 48.5594, 240.19, 75.3050, 219.43, 178.32, 361.8, 245.4445, 257.3, 451.04, 64.82, 417.03, 201.2229, 73.7, 70.6144
V1 , V4 , V6 , V8 , V10 , V12 , V15 , V18 , V19 , V24 , V25 , V26 , V27 , V31 , V32 , V34 , V36 , V40 , V42 , V46 , V49 , V54 , V55 , V56 , V59 , V61 , V62 , V65 , V66 , V69 , V70 , V72 , V73 , V74 , V76 , V77 , V80 , V85 , V87 , V89 , V90 , V91 , V92 , V99 , V100 , V103 , V104 , V105 , V107 , V110 , V111 , V112 , V113 , V116
0.9726, 1.0394, 0.9917, 1.0375, 1.0130, 0.9885, 1.0116, 1.0203, 1.0143, 1.0113, 1.0300, 1.0524, 1.0165, 0.9868, 1.0045, 1.0221, 1.0192, 0.9945, 0.9821, 1.0025, 1.0364, 1.0183, 1.0046, 1.0100, 1.0052, 0.9957, 1.0280, 1.0044, 1.0255, 1.0447, 1.0222, 1.0136, 1.0309, 0.9894, 0.9789, 1.0158, 1.0357, 0.9940, 1.0149, 0.9870, 0.9874, 0.9646, 0.9798, 1.0290, 1.0258, 1.0143, 0.9949, 0.9902, 0.9614, 0.9973, 1.0222, 0.9608, 1.0302, 1.0136
1.0147, 1.0354, 1.0196, 0.9611, 0.9503, 1.0138, 0.9922, 1.0095, 0.9943, 1.0355, 1.0558, 1.0279, 0.9904, 1.0105, 0.9960, 1.0072, 1.0141, 0.9698, 0.9788, 1.0030, 1.0244, 0.9938, 1.0155, 1.0017, 1.0401, 1.0487, 1.0324, 1.0381, 1.0169, 1.0543, 1.0396, 1.0285, 1.0136, 1.0088, 0.9888, 0.9887, 0.9868, 1.0127, 1.0301, 1.0408, 0.9724, 0.9825, 1.0116, 0.9884, 1.0239, 1.0335, 1.0118, 1.0091, 1.0157, 1.0248, 1.0231, 1.0369, 1.0208, 0.9868
TCSC (line outage 8–5) Fuel cost ($/h) Installation cost ($)
−0.5000, −0.4464, −0.3850 2.202 × 105 5.37 × 105
−0.4495, −0.1831, −0.2335 3.61 × 105 4.02 × 105
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Table 9 Best compromise solution for the IEEE 118-bus system. Control variables
Optimal control variable settings
P10 , P12 , P25 , P26 , P31 , P46 , P49 , P54 , P59 , P61 , P65 , P66 , P69 , P80 , P87 , P89 , P100 , P103 , P111
204.8711, 108.6736, 229.7154, 284.0657, 9.79, 28.55, 213.07, 63.30, 199.40, 156.40, 331.58, 319.44, 201.65, 405.52, 12.78, 215.64, 223.22, 35.70, 52.61
V1 , V4 , V6 , V8 , V10 , V12 , V15 , V18 , V19 , V24 , V25 , V26 , V27 , V31 , V32 , V34 , V36 , V40 , V42 , V46 , V49 , V54 , V55 , V56 , V59 , V61 , V62 , V65 , V66 , V69 , V70 , V72 , V73 , V74 , V76 , V77 , V80 , V85 , V87 , V89 , V90 , V91 , V92 , V99 , V100 , V103 , V104 , V105 , V107 , V110 , V111 , V112 , V113 , V116
1.0170, 0.9648, 0.9495, 1.0005, 0.9864, 1.0507, 0.9824, 1.0291, 1.0121, 0.9777, 1.0203, 1.0141, 1.0107, 0.9481, 0.9748, 1.0364, 0.9709, 0.9687, 1.0399, 1.0031, 0.9729, 1.0442, 1.0132, 1.0404, 0.9712, 0.9801, 0.9739, 0.9489, 0.9581, 0.9952, 1.0188, 0.9857, 1.0040, 1.0032, 0.9750, 0.9960, 0.9874, 0.9535, 0.9646, 0.9762, 1.0311, 1.0184, 1.0209, 0.9675, 0.9945, 0.9959, 1.0132, 1.0086, 0.9639, 0.9524, 0.9541, 0.9744, 1.0217, 1.0083
TCSC (line outage 8–5) Fuel cost ($/h) Installation cost ($)
−0.5000, −0.289, −0.4120 2.7402 × 105 4.38 × 105
7.3. Case 3: multi-objective SCOPF in IEEE 118-bus system In this case, the proposed algorithm was applied to alleviate the line overload under contingency condition in the IEEE 118-bus system. Contingency analysis was conducted on this system and the top three contingencies are produced in Table 6 along with the overloaded lines and the severity index value. LOSI values for the optimal location of TCSC are given in Table 7. Based on LOSI value, three locations were identified for the installation of TCSC to alleviate the line overload. In this case, the proposed algorithm was applied to enhance the security of the system for the first severe contingency. The best results of the MOGA were obtained with the following algorithm parameters: Number of generation: 120, population size: 50, cross over probability: 0.9 and mutation probability: 0.01. The pareto optimal front obtained for the first severe contingency case is plotted in Fig. 4. From the pareto front, two optimal solutions which are the extreme points of Fig. 4 that represents the best installation cost and best fuel cost are given in Table 8. The best compromise solutions are given in Table 9. Corresponding to this control variable it is found that there is no limit violation in any of the state variables in the base case and also in contingency cases. 8. Conclusion In this paper, the security enhancement task has been formulated as a multi-objective optimization problem and multiobjective genetic algorithm was applied to solve the same. The location of TCSC was identified based on Line Overload Sensitivity Index. It has considered as optimization criteria, the minimization of fuel cost and installation cost of TCSC. The algorithm has been tested on the standard IEEE 30-bus and IEEE 118-bus test systems. The proposed multi-objective GA has performed well when it was used to characterize pareto optimal front of the multiobjective optimal power flow problem. The MOGA emphasizes non-dominated solutions and simultaneously maintains diversity in the non-dominated solutions. In future, the proposed approach can be applied to solve security-constrained optimal power flow problems with multi-type FACTS devices. Appendix A. Severity index The severity of a contingency to line overload may be expressed in terms of the following severity index, which express the stress on the power system in the post contingency period:
severity index (SIC ) =
2m L0 Sl l=1
Slmax
(A.1)
where Sl = MVA flow in line ‘l’; Slmax = MVA rating of the line ‘l’. L0 = set of overloaded lines. m = integer exponent. Larger the severity index value a contingency has, the more severe it will be. The line flows in (A.1) are obtained from Newton–Raphson load flow calculations. While using the above severity index for security assessment, only the overloaded lines are considered to avoid masking effect.
References [1] O. Alsac, B. Scott, Optimal load flow with steady state security, IEEE Transactions on Power Systems PAS-93 (3) (1974) 745–751. [2] A. Monticelli, M.V.F. Pereira, S. Granville, Security constrained optimal power flow with post contingency corrective rescheduling, IEEE Transactions on Power Systems PWRS-2 (1) (1987) 175–182. [3] X. Duan, J. Chen, F. Peng, Y. Luo, Y. Huang, Power flow control with FACTS devices, IEEE Transactions on Power Systems 3 (2000) 1585–1589. [4] T.T. Lie, W. Deng, Optimal flexible AC transmission system (FACTS) devices allocation, Electrical Power & Energy Systems 19 (2) (1997) 125–134. [5] J. Mutale, Transmission network reinforcement versus FACTS: an economic assessment, IEEE Transactions on Power Systems 15 (3) (2000) 961–967. [6] G.D. Galiana, Assessment and control of the impact of FACTS devices on power system performance, IEEE Transactions on Power Systems 11 (4) (2000) 1931–1936. [7] K. Deb, Multi-objective Optimization Using Evolutionary Algorithms, John Wiley & Sons, Ltd., New York, 2005, pp. 209–213. [8] C.A.C. Coello, A.D. Christiansen, Moses, A multi-objective optimization tool for engineering design, Engineering Optimization 31 (3) (1999) 337–368. [9] C.S. Chang, K.P. Wong, B. Fan, Security-constrained multi-objective generation dispatch using bi-criterion global optimization, Proceedings of the Institute of Electrical and Electronics Engineers – Generation, Transmission & Distribution 142 (4) (1995) 406–414. [10] R. Yokoyama, S.H. Bae, T. Morita, H. Sasaki, Multi-objective generation dispatch based on probability security criteria, IEEE Transactions on Power Systems 3 (1) (1988) 317–324. [11] D. Radu, Y. Besanger, A multi-objective genetic algorithm approach to optimal allocation of multi-type FACTS devices for power systems security, in: Proceedings of the IEEE Power Engineering Society General Meeting, 2006, pp. 8–16. [12] K. Deb, A. Pratap, S. Agrawal, T. Meyarivan, A fast and elitist multi-objective genetic algorithm: NSGA-II, IEEE Transactions on Evolutionary Computation 6 (2) (2002) 182–197. [13] E. Zitzler, M. Laumanns, L. Thiele, SPEA2: improving the strength pareto evolutionary algorithm, in: Proceedings of Evolutionary Methods for Design, Optimization and Control with Applications to Industrial Problems – EUROGEN, 2001, pp. 95–100. [14] E. Zitzler, S. Kunzli, Indicator-based selection in multi-objective search, in: Proceedings of the Eighth International Conference on Parallel Problem Solving from Nature – PPSN VIII, 2004, pp. 832–842. [15] F. Xue, A.C. Sanderson, R.J. Graves, Pareto-based multi-objective differential evolution, in: Proceedings of the 2003 Congress on Evolutionary Computation – CEC’2003, vol. 2, Canberra, Australia, IEEE Press, 2003, pp. 862–869. [16] P.D. Justesen, R.K. Ursem, Many-objective distinct candidates optimization using differential evolution on centrifugal pump design problems, in: Proceedings of Evolutionary Computation (CEC), IEEE Congress, 2010, pp. 1–8. [17] L.J. Eshelman, J.D. Schaffer, in: D. Whitley (Ed.), Real-Coded Genetic Algorithms and Interval Schematam, 1993, pp. 187–202. [18] J.A. Wood, F.B. Woolenberg, Power Generation, Operation and Control, 2nd edition, Wiley India Pvt Ltd., New Delhi, 2006. [19] D.P. Kothari, J.S. Dhillon, Power System Optimization, 2nd edition, Prentice Hall of India Private Ltd, New Delhi, 2011, pp. 422–423. [20] IEEE 118-Bus System, Available at: www.ee.washintgton.edu, 1996.
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[21] E.-S. E.-S. El-Araby, N. Yorino, H. Sasaki, A two level hybrid GA/SLP for FACTS allocation problem considering voltage security, Electric Power and Energy Systems 25 (2003) 327–335. [22] D.E. Goldberg, Genetic Algorithms for Search, Optimization, and Machine Learning, Addison–Wesley, Reading, MA, 1989. [23] C.M. Fonseca, P.J. Fleming, An overview of evolutionary algorithms in multiobjective optimization, Evolutionary Computation 3 (1) (1995) 1–16. [24] D. Devaraj, Improved genetic algorithm for multi-objective reactive power dispatch problem, European Transactions on Electrical power 17 (6) (2007) 569–581. [25] P. Somasundaram, K. Kuppusamy, K. Devi, Evolutionary programming based security constrained optimal power flow, Electric Power System Research 72 (2004) 137–145. [26] C. Thitithamrongchai, B. EuaArpon, Security control optimal power flow: a parallel self-adaptive differential evolution approach, Electric Power Components and Systems 23 (10) (2005) 280–298. Dr. R. Narmatha Banu is an associate professor in Department of Electrical and Electronics Engineering, Kalasalingam University, Tamil Nadu, South India. She completed her B.E. (EEE) degree in Mohammed Sathak Engineering College, Kilakarai, Tamil Nadu, India in the year 1999 and M.E. (Power System Engg) degree in Annamalai University, Chidambaram, Tamil Nadu in the year 2002. She pursued Ph.D. in the Department of Electrical Engineering, Anna University, Chennai in the year 2010. Her area of interest is Power system Security, Genetic Algorithm and FACTS devices. She has got the Young Scientist Award for the year 2009 from Tamil Nadu State Council for Science and Technology (established by Government of Tamil Nadu).
Dr. D. Devaraj is a graduate from Thiagarajar College of Engineering in Electrical and Electronics Engineering (1992). He did his Masters in Power System Engineering from Madurai Kamaraj University, Madurai (1994). He obtained his Ph.D. from the Indian Institute of Technology, Chennai (2000) with specialization in power systems engineering. His research interests include power system engineering, power system automation, power system simulation, computational intelligent techniques, intelligent control techniques. He is currently working as a senior professor and Dean (research and development) in the Kalasalingam University, Krishnankoil, Tamil Nadu.
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Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc
Multi-objective GA with fuzzy decision making for security enhancement in power system R. Narmatha Banu ∗ , D. Devaraj Department of Electrical Engineering, Kalasalingam University, Krishnankoil, Tamil Nadu, India
a r t i c l e
i n f o
Article history: Received 18 January 2011 Received in revised form 10 August 2011 Accepted 7 March 2012 Available online 30 April 2012 Keywords: Power system security Flexible AC transmission system (FACTS) devices Thyristor Controlled Series Capacitors (TCSC) Genetic algorithm Pareto optimal frontier
a b s t r a c t Power system security enhancement is a major concern in the operation of power system. In this paper, the task of security enhancement is formulated as a multi-objective optimization problem with minimization of fuel cost and minimization of FACTS device investment cost as objectives. Generator active power, generator bus voltage magnitude and the reactance of Thyristor Controlled Series Capacitors (TCSC) are taken as the decision variables. The probable locations of TCSC are pre-selected based on the values of Line Overload Sensitivity Index (LOSI) calculated for each branch in the system. Multi-objective genetic algorithm (MOGA) is applied to solve this security optimization problem. In the proposed GA, the decision variables are represented as floating point numbers in the GA population. The MOGA emphasize non-dominated solutions and simultaneously maintains diversity in the non-dominated solutions. A fuzzy set theory-based approach is employed to obtain the best compromise solution over the trade-off curve. The proposed approach has been evaluated on the IEEE 30-bus and IEEE 118-bus test systems. Simulation results show the effectiveness of the proposed approach for solving the multi-objective security enhancement problem. © 2012 Elsevier B.V. All rights reserved.
1. Introduction In any power system, unexpected outage of transmission lines occurs due to faults or other disturbances. These events referred to as contingencies, may cause significant overloading of transmission lines, which in turn may lead to total or partial system blackout. Security constrained optimal power flow (SCOPF) is the main tool used in the energy control centers to avoid limit violation in the contingency state. SCOPF [1] adjusts base case decision variables to minimize the defined objective function subject to base case and contingency state operating constraints. The solution of an SCOPF is useful for both system operation and planning. The SCOPF does not take advantage of the post-contingency corrective rescheduling that is possible in static security enhancement. In [2], a mathematical framework was proposed for the solution of the SCOPF problem taking into account the system corrective capabilities such as generation rescheduling after the outage has occurred. The resulting dispatch has the same security level as the SCOPF, but with lower operating costs. An iterative approach is proposed in [6] to solve the SCOPF with corrective action.
∗ Corresponding author at: Kalasalingam University, Department of Electrical Engineering, Anandnagar, Krishnankoil 626126, Virudhunagar, Tamil Nadu, India. Tel.: +91 4563 289042; fax: +91 4563 289322. E-mail addresses: [email protected] (R.N. Banu), [email protected] (D. Devaraj). 1568-4946/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.asoc.2012.03.057
Apart from generation re-scheduling, FACTS devices [3,4] based on power electronics technology can also be used for power flow control through transmission lines. Thyristor Controlled Series Capacitors (TCSC), one of the FACTS devices can be used effectively in alleviating the line overload in case of a contingency. In this work, the base case generator active power, generator bus voltages and contingency state TCSC reactance values are used as the decision variables for security enhancement. For a large-scale power system, more than one FACTS device may have to be installed to achieve the desired performance. Studies have been conducted to identify the suitable location for FACTS devices to improve power system security. In this work, the location of TCSC is identified based on line overload severity index computed for every line in the system. While using FACTS devices for the performance improvement of power system, the installation cost need to be taken into account which is not done in the above papers. In this work, the installation cost of TCSC is taken as the additional objective of the OPF problem. In the literature, the SCOPF with corrective action is treated as a single-objective optimization problem [5,6]. In this paper, the SCOPF with corrective action is treated as a true multi-objective optimization problem with minimization of fuel cost and the installation cost of TCSC as the objectives. Because of the presence of conflicting multiple objectives, a multi-objective optimization problem results in a number of optimal solutions, known as pareto optimal solutions [7,8]. In a multi-objective optimization, effort must be made in finding the set of trade-off optimal solutions by considering all objectives to be important.
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Gij , Bij Gii , Bii Gk FT NB NB−1 NPQ Ng Nl Pi , Qi Pgi , Qgi Sl Slmax Vi Vj ij
mutual conductance and susceptance between bus i and bus j self-conductance and susceptance of bus i conductance of branch k total fuel cost total number of buses total number of buses excluding slack bus number of PQ buses number of generator buses number of branches in the system real and reactive powers injected into network at bus i real and reactive power generation at bus i apparent power flow through the lth branch apparent power flow limit through the lth branch voltage magnitude at bus i voltage magnitude at bus j voltage angle difference between bus i and bus j
One, straightforward approach to solve the multi-objective optimization problem is to convert them into a single objective problem by linear combination of different objectives as a weighted sum and then solve it similar to single objective optimization problems [9]. The important aspect of this weighted sum method is that a set of non-inferior (or pareto optimal) solutions can be obtained by varying the weights. Unfortunately, this requires multiple run as many times as the number of desired pareto optimal solutions. Furthermore, this method cannot be used to find pareto optimal solutions in problems having a non-convex pareto optimal front. To avoid this difficulty, the Є-constraint method [10] is used for multi-objective optimization problem. This method is based on optimizing the most preferred objective and considering the other objectives as constraints bounded by some allowable levels. These levels are then altered to generate the entire pareto optimal set. This approach is time-consuming and tends to find weak pareto optimal solutions. The ability of Evolutionary Computation techniques like Genetic Algorithm to find multiple optimal solutions in one single simulation run makes them unique in solving multi-objective optimization problems [7]. In this work, the multi-objective security optimization problem is solved using multi-objective genetic algorithm (MOGA) [11]. Like the other approaches such as NSGA II, SPEA2, IBEA and DEMO, MOGA is also a population-based search algorithm. Each algorithm differs in the way fitness value is assigned to the individuals while solving the multi-objective optimization problem. The environmental selection in NSGA-II [12] first ranks the individuals using non-dominated sorting. To distinguish between individuals with the same rank, the crowding distance metric is used, which prefers individuals from less crowded regions of the objective space. SPEA2 [13] works similarly, calculating the raw fitness of the individuals according to Pareto dominance relations between them and using a density measure to break the ties. The individuals that reside close together in the objective space are discouraged from entering the archive of best solutions. IBEA [14], on the other hand, uses a different approach. The fitness of individuals is determined only according to the value of a predefined indicator. This indicator has to be dominance preserving and no other explicit diversity preserving mechanism (such as crowding in NSGA-II or density in SPEA2) is applied. In DEMO (Differential Evolution for Multi-objective Optimization) [15], the fitness of an individual is first calculated using Pareto-based ranking and then reduced with respect to the individual’s crowding distance value. This single
-jxc
i
Nomenclature
2757
rij+jxij
j
Fig. 1. Equivalent circuit of TCSC.
fitness value is then used to select the best individuals for the new population. Generally, binary strings are used to represent the decision variables of the optimization problem in the genetic population irrespective of the nature of the decision variables. This binarycoded GA has Hamming cliff problems [17] which sometimes may cause difficulties in the case of coding continuous variables. Also, for discrete variables with total number of permissible choices not equal to 2k (where k is an integer) it becomes difficult to use a fixed length binary coding to represent all permissible values. To overcome these difficulties, in this paper, the optimization variables namely, generator active power generation Pgi , generator bus voltages Vgi and TCSC settings are represented as floating point numbers in the genetic population. For effective genetic operation, crossover and mutation operators which can directly operate on floating point numbers [24] are used. The effectiveness and potential of the proposed approach to solve the multi-objective optimal power flow (OPF) problem has been demonstrated using IEEE 30bus and IEEE 118-bus systems. Lesser computational time taken by the MOGA to reach the optimal solutions makes it suitable for solving the large scale optimization problem like SCOPF. 2. Modelling and placement of Thyristor Controlled Series Capacitors (TCSC) Thyristor Controlled Series Capacitors (TCSC) consist of a fixed capacitor in parallel with a thyristor controlled reactor. The primary function of the TCSC is to provide variable series compensation to a transmission line. This changes the line flow due to change in series reactance. Fig. 1 shows a model of transmission line with TCSC connected between buses ‘i’ and ‘j’. For steady state analysis, the TCSC can be considered as a static reactance −jxc . The controllable reactance xc is directly used as the control variable in the power flow equations. The power flow equations of a transmission line with TCSC can be written as: Pij = Vi2 gij − Vi Vj (gij cos ıij + bij sin ıij ) Qij = −Vi2 bij − Vi Vj (gij sin ıij − bij cos ıij )
(1)
where rij
gij =
rij2 + (xij − xc )2 xc bij = xij − rij2 + (xij − xc )2
(2)
The only difference between normal line power flow equation and the TCSC line power flow equation is the presence of the controllable reactance xc which is varied by adjusting the value of TCSC reactance. To enhance the security of the system, the TCSC has to be placed at the suitable locations. To determine the best location of TCSC, an index called Line Overload Sensitivity Index (LOSI) is calculated for all the remaining lines. The LOSIl for branch “l” is defined as the sum of the normalized power flow through branch “l” to all the considered contingencies ‘C’, expressed as: LOSIl =
NC SlC C=1
Slmax
(3)
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where SlC = MVA flow in line ‘l’ during contingency “C”. The LOSI defined at branch “l” for the base case loading is defined by LOSIlBL . In order to achieve optimal location of TCSC, valid under change in system loading, LOSI indices defined in (4) are also computed at an increased loading and decreased loading scenario. The increased loading scenario pertains to all the loads increased by 5% from their base values and the decreased loading scenario has been simulated with the loads decreased by 5% from their base values. The corresponding LOSI, calculated at each overloaded lines, are termed as LOSIlIL and LOSIlDL respectively. The optimal location of TCSC has been decided by an average line overload severity index, computed for every line, as defined in the following:
LOSIl =
LOSIlBL + LOSIlIL + LOSIlDL 3
(4)
The branches are ranked based on their corresponding LOSIl values. The TCSC are placed on the branches starting from the top of the ranking list and proceeding downward with as many branches as the number of available TCSC. 3. Problem formulation
3.1. Objective functions 3.1.1. Economic objective function (FC ) The most commonly used objective in the OPF problem formulation is the minimization of the total operation cost of the fuel consumed for producing electric power within a scheduled time interval. The individual costs of each generating unit are assumed to be function only of real power generation and are represented by quadratic curves. The objective function for the entire power system can be expressed as the sum of the quadratic cost model at each generator [18]. Ng
2 (ai Pgi + bi Pgi + ci ) ($/h)
(5)
i=1
where ai , bi , and ci are the cost coefficients of generator at bus i. Pgi is the active power generation at bus i. 3.1.2. TCSC cost function (FE ) It is important to take the economical aspects of the TCSC devices installed in the power system due to high investment and operating costs. The total investment cost of the TCSC device is expressed as [21]: FIC =
(fi + vi Xi )
• load flow constraints Qi − Vi
NB
Vj (Gij sin ij − Bij cos ij ) = 0,
(6)
i∈˝
where fi and vi are the fixed cost and variable cost for candidate. ˝ is a set of all candidate sites, and Xi , is the rating of TCSC device i.
i = 1, 2, . . . , NPQ
j=1
(7)
Pi − Vi
NB
Vj (Gij cos ij + Bij sin ij ) = 0,
i = 1, 2, . . . , NB−1
(8)
j=1
• voltage constraint Vimin ≤ Vi ≤ Vimax ,
i ∈ NB
(9)
• real power generation limit Pgmin ≤ Pg ≤ Pgmax ,
g ∈ Ng
(10)
• generator reactive power generation limit Qgmin ≤ Qg ≤ Qgmax ,
In general, the optimal power flow (OPF) problem is formulated as an optimization problem in which one or more objective functions are minimized while satisfying a number of equality and inequality constraints. In the security enhancement problem considered here the goal is to determine the optimal values of generator active power, generator bus voltage magnitudes and TCSC that enhance the systems security level while minimizing the generator fuel cost and investment cost of TCSC. Minimization of fuel cost will necessitate higher values of TCSC to reach the same level of security. This will be lead to increased value of TCSC installation cost. The mathematical formulation of the security enhancement problem is given below.
FC =
3.1.3. Problem constraints The constraints are:
g ∈ Ng
(11)
• transmission line flow limit Sl < Slmax ,
l ∈ Nl
(12)
• limit on reactance of TCSC min max ≤ XTCSC,i ≤ XTCSC,i , XTCSC,i
i ∈ NTCSC
(13)
The security constrained optimal power flow (SCOPF) formulation considers both the pre and post contingency state power flows and all constraints in those states should always be satisfied. The post-contingency constraints are of the same dimension as those of the pre-contingency case. If there are m total constraints in a given base case optimal power flow (OPF), there will be (k + 1)m constraints in a SCOPF formulation with k contingencies. Aggregating the objectives and constraints, the problem can be mathematically formulated as a non-linear constrained multiobjective optimization problem as follows: minimize FT = [FC , FIC ]
(14)
subject to the constraints (7)–(13). 4. Multi-objective genetic algorithm Genetic algorithms (GA) [22] are generalized search algorithms based on the mechanics of natural genetics. GA maintains a population of individuals that represent the candidate solutions to the given problem. Each individual in the population is evaluated to give some measure of its fitness to the problem from the objective function. GAs combine solution evaluation with stochastic genetic operators namely, selection, crossover and mutation to obtain near optimality. Being a population-based approach, GA is well suited to solve multi-objective optimization problems. Multi-objective genetic algorithm [7,23] is an extension of classical GA. The main difference between a conventional GA and a MOGA lies in the assignment of fitness to an individual. The rest of the algorithm is the same as that in a classical GA. The details of the MOGA are described below. In the MOGA, first, each solution is checked for its domination in the population. Two solutions (x(1) and x(2) ) are compared on the basis of whether one dominates the other solution or not. A solution x(1) is said to dominate the other solution x(2) , if the following conditions are satisfied:
R.N. Banu, D. Devaraj / Applied Soft Computing 12 (2012) 2756–2764
(a) The solution x(1) is no worse than x(2) in all objectives, or ¯ i (x(2) ) for all i = 1, 2, . . ., M where M be the objective fi (x(1) )f functions. (b) The solutions x(1) is strictly better than x(2) in at least one objective, or fi (x(1) ) fi (x(2) ) for at least one j (j ∈ {1, 2, . . ., M}) If any of the above condition is violated, the solution x(1) does not dominate the solution x(2) (or mathematically x(1) ≤ x(2) ). To a solution ‘i’, a rank ri equal to one plus the number of solutions i that dominate solution ‘i’ is assigned: ri = 1 + i
(15)
In this way, non-dominated solutions are assigned a rank equal to 1, since no solution would dominate a non-dominated solution in a population. Once the ranking is done, a raw fitness is assigned to each solution based on its rank. To perform this, first the ranks are sorted in ascending order of magnitude. Then, a raw fitness is assigned to each solution by using a mapping function. Usually, the mapping function is chosen so as to assign fitness between N (for the best rank solution) and 1 (for the worst rank solution). Thereafter, solutions of each rank are considered at a time and their raw fitness are averaged. This average fitness is called the assigned fitness to each solution of the rank. This process emphasizes non-dominated solutions in the population. In order to maintain diversity among non-dominated solutions, niching among solutions of each rank are introduced. The niche count is calculated with the following equation: nci =
(ri )
Sh(dij )
(16)
j=1
where (ri ) is the number of solutions in rank ri and Sh(dij ) is the sharing function of two solutions i and j. The sharing function Sh(d) is calculated using objective function value as distance metric as Sh(dij ) =
⎧ ⎨ ⎩
1−
˛
dij share
,
0,
if d ≤ share
k=1
j
fki − fk fkmax − fkmin
2 (18)
where fkmax and fkmin are the maximum and minimum objective function value of the kth objective. The shared function takes a value in [0, 1], depending on the values of dij and share . The shared fitness value is calculated by dividing the assigned fitness of a solution by its niche count. Although all solutions of any particular rank have the identical fitness, the shared fitness value of a solution residing in a less crowded region has a better shared fitness. This produces a large selection pressure for poorly represented solutions in any rank. Dividing the assigned fitness value by the niche count reduces the fitness of each solution. In order to keep the average fitness of the solutions in a rank the same as that before sharing, these fitness values are scaled using Eq. (19) so that their average shared fitness value is the same as the average assigned fitness value. fjSC =
fj (r)
(r)
f k=1 k
culated using fj = fj /ncj ; (r) is the number of solutions in rank ri . This procedure is continued until all ranks are processed. Thereafter, selection, crossover and mutation operators are applied to create a new population. With each individual represented as a string of integers and floating point numbers, selection process remains the same as in classical GA, but the cross over and mutation operators are applied variable by variable. In this paper, tournament selection and BLX-␣ crossover and non-uniform mutation operators [7] are used.
5. Best compromise solution Upon having the pareto optimal set of non-dominated solution, it is preferred to get the best compromise solution for implementation. The Many-objective Distinct Candidates Optimization using Differential Evolution (MODCODE) algorithm [16] discovered a low number of solutions within a region of interest on the true pareto front. It aims at returning a few optimal distinct solutions within a region of interest, with both result set cardinality and distinctiveness being user defined in compliance with the goals of MODCO algorithm. Considering the imprecise nature of decision maker’s judgment this work applies a fuzzy set-based approach to obtain the best compromise solution [19]. Fuzzy set theory generalizes classical set theory to allow partial membership with a smooth boundary. The degree of membership in a set is expressed by a number between 0 and 1.0 means entirely not in the set, 1 means completely in the set, and a number in between 0 and 1 means partially in the set. In this paper, it is assumed that the decision maker (DM) has fuzzy goal for each objective. These fuzzy goals depend on experience and intuitive knowledge of the DM. The fuzzy goals are represented by linear membership function as given by Eq. (20). The ith objective function Fi is represented by a membership function i defined by
i (Fi ) =
where ‘ share ’ is the sharing parameter which signifies the maximum distance between any two solutions before they can be considered to be in the same niche and dij is the normalized distance between any two solutions i and j in a rank. The normalized distance dij is calculated using
M dij =
where fjSC is the scaled fitness; fj is the shared fitness and it is cal-
(17)
otherwise
(19)
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⎧ 1, ⎪ ⎨ F max − F
i
i
max min ⎪ ⎩ Fi − Fi
0,
Fi ≤ Fimin , Fimin < Fi < Fimax Fi ≥
(20)
Fimax
where Fimin and Fimax are the minimum and maximum value of the ith objective function respectively among all non-dominated solutions. The value of membership function suggests how far (in the scale from 0 to 1) a non-inferior (non-dominated) solution has satisfied the Fi objective. The sum of membership function values (Fi ) (i = 1, 2, 3, . . ., M) for all the objectives can be computed in order to measure the accomplishment of each solution in satisfying the objectives. The accomplishment of each non-dominated solutions can be rated with respect to all the N non-dominated solutions by normalizing its accomplishment over the sum of the accomplishments of N non-dominated solutions as follows:
M
k
=
i=1
(Fik )
N M k=1
i=1
(Fik )
(21)
The solution that attains the maximum membership k , in the fuzzy set so obtained can be chosen as the best solution or the one having the highest cardinal priority ranking. 6. Genetic algorithm implementation While applying GA for solving the SCOPF problem, the following issues need to be addressed:
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Table 1 Line outage ranking using severity index. Outage line no.
Over loaded lines
Line flow (MVA)
Line flow limit (MVA)
Severity index (SI)
Rank
1–2
1–3 3–4 4–6
191.58 174.13 103.37
130 130 90
5.262
1
1–3
1–2 2–6
181.17 66.482
130 65
3.010
2
3–4
1–2 2–6
178.43 65.558
130 65
2.9011
3
22–24 24–25
19.062 17.781
16 16
2.1979
4
2–6 5–7
76.285 101.08
65 70
1.3777
5
28–27 2–5
• solution representation and • fitness evaluation.
⎧ ⎨ KV (ViK − Vimax )2 , if Vik > Vimax
PViK =
2
k K min min ⎩ KV (Vi − Vi ) , if Vi < Vi
0,
6.1. Solution representation Implementation of GA for a problem starts with the parameter encoding (i.e., the representation of the problem). Each individual in the genetic population represents a candidate solution. The elements of that solution consist of all the decision variables in the system. The decision variables of the SCOPF problem include active power generation Pgi , generator bus voltage magnitude Vgi and transformer tap settings tk . The solution variables are represented as floating point numbers and integers. This representation has a number of advantages over binary coding. The efficiency of the GA is increased as there is no need to convert the solution variables to the binary type. Also, the computer memory required to store the population is reduced.
PQiK =
otherwise
⎧ ⎨ Kq (QiK − Qimax )2 , if Qik > Qimax 2
k K min min ⎩ Kq (Qi − Qi ) , if Qi < Qi
PLlK =
(24)
0,
(25)
otherwise
2
Kl (SlK − Slmax ) , 0,
if Slk > Slmax otherwise
(26)
where Ks , Kv , Kq , and Kl are the penalty factors for slack bus power output, bus voltage limit violation, generator reactive power limit violation and line flow violation respectively. The penalty function is added to each of the objective function to get the new objective functions. 7. Simulation results
6.2. Evaluation function GA searches for the optimal solution by maximizing a given fitness function, and therefore an evaluation function which provides a measure of the quality of the problem solution must be provided. In the SCOPF problem under consideration, the objectives are to minimize the fuel cost of generation and investment cost of TCSC satisfying the constraints. The equality constraints given by Eqs. (7) and (8) are satisfied by running the power flow program. The active power generation (Pgi ) (except the generator at the slack bus), generator terminal bus voltages (Vgi ), transformer tap settings (tk ) and reactance of TCSC (XTCSC ) are the decision variables and they are self-restricted by the optimization algorithm. The limit on active power generation at the slack bus (Pgs ), load bus voltages (Vload ), reactive power generation (Qgi ) and line flow (Sl ) are satisfied by adding a penalty function with the objective function. With the inclusion of the penalty terms, the over all penalty function becomes: PF =
NC
PS K +
K=0
NC NPQ
PViK +
K=0 i=1
NC Ng
PQiK +
K=0 i=1
NC Nl
PLlK
(22)
K=0 l=1
where PSK , PViK , PQiK and PLlK are the penalty terms for the slack bus generator active power limit violation, load bus voltage-limit violation, reactive power generation limit violation, and line flow limit violation respectively. These quantities are defined by the equations:
PS K =
⎧ ⎨ KS (PSK − PSmax )2 , if PSK > PSmax 2
K min K min ⎩ KS (PS − PS ) , if PS < PS
0,
otherwise
(23)
The proposed multi-objective genetic algorithm approach has been applied to solve the security enhancement problem in IEEE30 bus and IEEE 118-bus test systems. The IEEE 30-bus system has 6 generator buses, 24 load buses and 41 transmission lines, of which 4 branches (6–9), (6–10), (4–12) and (28–27) are with tap setting transformers. The generator and transmission-line data relevant to the system are taken from [1]. The upper and lower voltage limits at all the bus bars except slack bus are taken as 1.10 p.u. and 0.95 p.u. respectively. The slack bus bar voltage is fixed to its specified value of 1.06 p.u. The generator cost coefficients and the transmission line parameters are taken from [1]. The IEEE 118-bus test system has 54 generator buses and 186 transmission lines. All other data are the same as the standard IEEE 118-bus data given in [20]. To demonstrate the effectiveness of the proposed approach, three different cases have been considered as follows: Case 1: SCOPF with minimization of fuel cost as objective in IEEE 30-bus system without considering post contingency rescheduling. Case 2: Multi objective optimal power flow for security enhancement in IEEE 30-bus system. Case 3: Multi objective optimal power flow for security enhancement in IEEE 118-bus system. 7.1. Case 1: SCOPF with minimization of fuel cost as objective in IEEE 30-bus system In this case, contingency analysis was carried out on the system to identify the severe contingencies. The list of severe contingencies along with their severity index value which were from Eq.
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Table 4 Extreme solutions for the three severe contingencies. Control variables
Minimum fuel cost solution
Minimum installation cost solution
P1 P2 P5 P8 P11 P13 V1 V2 V5 V8 V11 V13 TCSC (line outage 1–2)
124.92 58.64 25.32 34.89 22.934 20.67 1.0489 1.0345 1.0096 1.0147 1.0986 1.0580 −0.5000, −0.0734 −0.5000, −0.2441 −0.4173, −0.4336 814.7269 3.56 × 104
129.74 44.121 40.507 27.471 25.984 17.508 1.0476 1.0322 1.002 1.0023 1.0597 1.0566 −0.5000, −0.4958 −0.3468, −0.4709 −0.2895, −0.2612 827.5058 0.47 × 104
TCSC (line outage 1–3) TCSC3 (line outage 3–4)
Fig. 2. Convergence of the GA-SCOPF algorithm for IEEE 30-bus test system.
(A.1) is given in Table 1. From the contingency analysis, it is found that five contingencies have resulted in overload on other lines. The line flows corresponding to the severe contingencies were included as additional constraints of the OPF problem and the proposed GA was applied to solve this security constrained OPF problem with the minimization of base case fuel cost as the objective function. Generator active power outputs and the generator bus terminal voltages were taken as the optimization variables. The optimization variables are represented as floating point numbers in the GA population. The initial population was randomly generated between the variable’s lower and upper limits. Tournament selection was applied to select the members of the new population. Blend crossover and uniform mutation were applied on the selected individuals. The performance of GA generally depends on the GA parameters used, in particular, the crossover and mutation probabilities, Pc and Pm, respectively. The performance of GA for various crossover and mutation probabilities in the range of 0.6–0.9 and 0.001–0.01 respectively was therefore evaluated. The best result of the GA was obtained with the following algorithm parameters:
Fuel cost ($/h) Installation cost ($)
Table 5 Best compromise solution for the IEEE 30-bus system. Control variables
Optimal Control variable settings
P1 , P2 , P5 , P8 , P11 , P13 (base)
125.265,58.645, 25.317, 34.89, 22.93, 20.67 0.9750, 1.0500, 1.1000, 0.9800, 0.9670, 0.9520, 1.0140, 1.0750, 0.9840. −0.5000, −0.0404 −0.4900, −0.321 −0.323, −0.4321 819.41 1.52 × 104
V1 , V2 , V5 , V8 , V11 , V13 TCSC (line outage 1–2) TCSC (line outage 1–3) TCSC3 (line outage 3–4) Fcost ($/h) Installation cost ($)
No. of generations: 100, population size: 50, crossover probability: 0.9, mutation probability: 0.01. After 100 generations it was found that all the individuals have reached almost the same fitness value. This shows that GA has reached the optimal solution. Fig. 2 shows the variation of fitness during the GA run for the best case. Ten trail runs of the GA was conducted for the SCOPF problem and the minimum, maximum and the average value of fuel cost are 824.8991 $/h, 824.991 $/h and 824.9231 $/h respectively.
Table 2 Result of SCOPF Algorithm. P1 P2 P5 P8 P11 P13 V1 V2 V5 V8 V11 V13
142.38 49.18 29.51 34.59 16.70 20.6 1.0498 1.0340 1.0105 1.0245 0.9310 1.0264
Generation cost
824.9868 ($/h)
Table 3 LOSI values for IEEE 30 bus test system. S. no.
Branches
LOSI value
1 2
2–6 2–4
0.93942 0.3283
Fig. 3. Pareto-optimal front for the three severe contingencies.
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Table 6 Line outage ranking using severity index for IEEE 118-bus system. Outage line no.
Over loaded lines
Line flow (MVA)
Line flow limit (MVA)
Severity index (SI)
Rank
8–5
30–17 8–30
506.554 419.466
500 200
5.4252
1
9–10
8–30 30–38
214.225 198.765
200 175
3.1158
2
8–9
8–30 30–38
270.172 198.713
200 175
3.1142
3
The minimum cost obtained by the GA based approach along with the optimal decision variables are given in Table 2. The algorithm took 129 s to reach the optimal solution. Corresponding to these decision variables, it was found that all the state variables satisfy the lower and upper limits. The secured optimal solution obtained by this algorithm does not violate any constraints, whereas the optimal solution reported in [1] violates line loading limit under contingencies 1–2 and 28–27 and the results given in [25] violates some line loading limit under contingency 1–2. Further, the fuel cost obtained by the proposed algorithm is less than the value reported in [26] for the same IEEE 30-bus system. 7.2. Case 2: multi-objective SCOPF in IEEE 30-bus system In this case, the multi-objective genetic algorithm was applied to enhance the security of the system for the first three contingencies. The LOSI values are calculated using (3) for each branch of the studied system for the severe contingencies. The top two branches which posses high values of LOSI are listed in Table 3. The TCSC are placed in these two lines. It is assumed that the impedance of all TCSC can be varied within 50% of the corresponding branch impedance. Generator active power, generator bus bar voltages and the reactance values of TCSC are taken as the decision variables and the problem was handled as a multi-objective optimization problem with TCSC installation cost and generator fuel cost as objectives to be minimized simultaneously. For the considered power system, the MOGA was applied considering several sets of parameters in order to prove its capability to provide acceptable trade-offs close to the Pareto Optimal Front (POF). In all simulation, the following parameters were used: Number of generation: 100, population size: 50, cross over probability: 0.9 and mutation probability: 0.01. The pareto optimal front obtained for the three contingency cases are plotted in Fig. 3. It is worth mentioning that the proposed approach has produced 16 pareto optimal solutions in a single run
Table 7 LOSI values for IEEE 118-bus test system. S. no.
Branches
LOSI value
1 2 3
8–30 30–38 30–17
1.2208 0.8358 0.7199
that have satisfactory diversity characteristics and span the entire pareto optimal front. Out of these, two optimal solutions which are the extreme points of Fig. 3 that represent the best installation cost and best fuel cost are given in Table 4. The best compromise solutions which were found from Eqs. (20) and (21) are given in Table 5. In all cases, the value of SI is zero, which shows that the proposed approach is able to alleviate the line overload.
Fig. 4. Pareto-optimal front for the severe contingency.
Table 8 Pareto optimal solutions for the most severe contingencies. Control variables
Minimum fuel cost solution
Minimum installation cost solution
P10 , P12 , P25 , P26 , P31 , P46 , P49 , P54 , P59 , P61 , P65 , P66 , P69 , P80 , P87 , P89 , P100 , P103 , P111
201.8711, 108.6736, 219.7154, 264.0657, 19.7902, 28.5594, 213.0799, 63.3050, 199.4073, 156.4032, 331.5864, 231.4445, 201.6571, 405.5204, 12.7802, 215.6403, 223.2229, 35.7076, 52.6144
220.8711, 118.76, 148.7154, 283.0657, 53.87, 48.5594, 240.19, 75.3050, 219.43, 178.32, 361.8, 245.4445, 257.3, 451.04, 64.82, 417.03, 201.2229, 73.7, 70.6144
V1 , V4 , V6 , V8 , V10 , V12 , V15 , V18 , V19 , V24 , V25 , V26 , V27 , V31 , V32 , V34 , V36 , V40 , V42 , V46 , V49 , V54 , V55 , V56 , V59 , V61 , V62 , V65 , V66 , V69 , V70 , V72 , V73 , V74 , V76 , V77 , V80 , V85 , V87 , V89 , V90 , V91 , V92 , V99 , V100 , V103 , V104 , V105 , V107 , V110 , V111 , V112 , V113 , V116
0.9726, 1.0394, 0.9917, 1.0375, 1.0130, 0.9885, 1.0116, 1.0203, 1.0143, 1.0113, 1.0300, 1.0524, 1.0165, 0.9868, 1.0045, 1.0221, 1.0192, 0.9945, 0.9821, 1.0025, 1.0364, 1.0183, 1.0046, 1.0100, 1.0052, 0.9957, 1.0280, 1.0044, 1.0255, 1.0447, 1.0222, 1.0136, 1.0309, 0.9894, 0.9789, 1.0158, 1.0357, 0.9940, 1.0149, 0.9870, 0.9874, 0.9646, 0.9798, 1.0290, 1.0258, 1.0143, 0.9949, 0.9902, 0.9614, 0.9973, 1.0222, 0.9608, 1.0302, 1.0136
1.0147, 1.0354, 1.0196, 0.9611, 0.9503, 1.0138, 0.9922, 1.0095, 0.9943, 1.0355, 1.0558, 1.0279, 0.9904, 1.0105, 0.9960, 1.0072, 1.0141, 0.9698, 0.9788, 1.0030, 1.0244, 0.9938, 1.0155, 1.0017, 1.0401, 1.0487, 1.0324, 1.0381, 1.0169, 1.0543, 1.0396, 1.0285, 1.0136, 1.0088, 0.9888, 0.9887, 0.9868, 1.0127, 1.0301, 1.0408, 0.9724, 0.9825, 1.0116, 0.9884, 1.0239, 1.0335, 1.0118, 1.0091, 1.0157, 1.0248, 1.0231, 1.0369, 1.0208, 0.9868
TCSC (line outage 8–5) Fuel cost ($/h) Installation cost ($)
−0.5000, −0.4464, −0.3850 2.202 × 105 5.37 × 105
−0.4495, −0.1831, −0.2335 3.61 × 105 4.02 × 105
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Table 9 Best compromise solution for the IEEE 118-bus system. Control variables
Optimal control variable settings
P10 , P12 , P25 , P26 , P31 , P46 , P49 , P54 , P59 , P61 , P65 , P66 , P69 , P80 , P87 , P89 , P100 , P103 , P111
204.8711, 108.6736, 229.7154, 284.0657, 9.79, 28.55, 213.07, 63.30, 199.40, 156.40, 331.58, 319.44, 201.65, 405.52, 12.78, 215.64, 223.22, 35.70, 52.61
V1 , V4 , V6 , V8 , V10 , V12 , V15 , V18 , V19 , V24 , V25 , V26 , V27 , V31 , V32 , V34 , V36 , V40 , V42 , V46 , V49 , V54 , V55 , V56 , V59 , V61 , V62 , V65 , V66 , V69 , V70 , V72 , V73 , V74 , V76 , V77 , V80 , V85 , V87 , V89 , V90 , V91 , V92 , V99 , V100 , V103 , V104 , V105 , V107 , V110 , V111 , V112 , V113 , V116
1.0170, 0.9648, 0.9495, 1.0005, 0.9864, 1.0507, 0.9824, 1.0291, 1.0121, 0.9777, 1.0203, 1.0141, 1.0107, 0.9481, 0.9748, 1.0364, 0.9709, 0.9687, 1.0399, 1.0031, 0.9729, 1.0442, 1.0132, 1.0404, 0.9712, 0.9801, 0.9739, 0.9489, 0.9581, 0.9952, 1.0188, 0.9857, 1.0040, 1.0032, 0.9750, 0.9960, 0.9874, 0.9535, 0.9646, 0.9762, 1.0311, 1.0184, 1.0209, 0.9675, 0.9945, 0.9959, 1.0132, 1.0086, 0.9639, 0.9524, 0.9541, 0.9744, 1.0217, 1.0083
TCSC (line outage 8–5) Fuel cost ($/h) Installation cost ($)
−0.5000, −0.289, −0.4120 2.7402 × 105 4.38 × 105
7.3. Case 3: multi-objective SCOPF in IEEE 118-bus system In this case, the proposed algorithm was applied to alleviate the line overload under contingency condition in the IEEE 118-bus system. Contingency analysis was conducted on this system and the top three contingencies are produced in Table 6 along with the overloaded lines and the severity index value. LOSI values for the optimal location of TCSC are given in Table 7. Based on LOSI value, three locations were identified for the installation of TCSC to alleviate the line overload. In this case, the proposed algorithm was applied to enhance the security of the system for the first severe contingency. The best results of the MOGA were obtained with the following algorithm parameters: Number of generation: 120, population size: 50, cross over probability: 0.9 and mutation probability: 0.01. The pareto optimal front obtained for the first severe contingency case is plotted in Fig. 4. From the pareto front, two optimal solutions which are the extreme points of Fig. 4 that represents the best installation cost and best fuel cost are given in Table 8. The best compromise solutions are given in Table 9. Corresponding to this control variable it is found that there is no limit violation in any of the state variables in the base case and also in contingency cases. 8. Conclusion In this paper, the security enhancement task has been formulated as a multi-objective optimization problem and multiobjective genetic algorithm was applied to solve the same. The location of TCSC was identified based on Line Overload Sensitivity Index. It has considered as optimization criteria, the minimization of fuel cost and installation cost of TCSC. The algorithm has been tested on the standard IEEE 30-bus and IEEE 118-bus test systems. The proposed multi-objective GA has performed well when it was used to characterize pareto optimal front of the multiobjective optimal power flow problem. The MOGA emphasizes non-dominated solutions and simultaneously maintains diversity in the non-dominated solutions. In future, the proposed approach can be applied to solve security-constrained optimal power flow problems with multi-type FACTS devices. Appendix A. Severity index The severity of a contingency to line overload may be expressed in terms of the following severity index, which express the stress on the power system in the post contingency period:
severity index (SIC ) =
2m L0 Sl l=1
Slmax
(A.1)
where Sl = MVA flow in line ‘l’; Slmax = MVA rating of the line ‘l’. L0 = set of overloaded lines. m = integer exponent. Larger the severity index value a contingency has, the more severe it will be. The line flows in (A.1) are obtained from Newton–Raphson load flow calculations. While using the above severity index for security assessment, only the overloaded lines are considered to avoid masking effect.
References [1] O. Alsac, B. Scott, Optimal load flow with steady state security, IEEE Transactions on Power Systems PAS-93 (3) (1974) 745–751. [2] A. Monticelli, M.V.F. Pereira, S. Granville, Security constrained optimal power flow with post contingency corrective rescheduling, IEEE Transactions on Power Systems PWRS-2 (1) (1987) 175–182. [3] X. Duan, J. Chen, F. Peng, Y. Luo, Y. Huang, Power flow control with FACTS devices, IEEE Transactions on Power Systems 3 (2000) 1585–1589. [4] T.T. Lie, W. Deng, Optimal flexible AC transmission system (FACTS) devices allocation, Electrical Power & Energy Systems 19 (2) (1997) 125–134. [5] J. Mutale, Transmission network reinforcement versus FACTS: an economic assessment, IEEE Transactions on Power Systems 15 (3) (2000) 961–967. [6] G.D. Galiana, Assessment and control of the impact of FACTS devices on power system performance, IEEE Transactions on Power Systems 11 (4) (2000) 1931–1936. [7] K. Deb, Multi-objective Optimization Using Evolutionary Algorithms, John Wiley & Sons, Ltd., New York, 2005, pp. 209–213. [8] C.A.C. Coello, A.D. Christiansen, Moses, A multi-objective optimization tool for engineering design, Engineering Optimization 31 (3) (1999) 337–368. [9] C.S. Chang, K.P. Wong, B. Fan, Security-constrained multi-objective generation dispatch using bi-criterion global optimization, Proceedings of the Institute of Electrical and Electronics Engineers – Generation, Transmission & Distribution 142 (4) (1995) 406–414. [10] R. Yokoyama, S.H. Bae, T. Morita, H. Sasaki, Multi-objective generation dispatch based on probability security criteria, IEEE Transactions on Power Systems 3 (1) (1988) 317–324. [11] D. Radu, Y. Besanger, A multi-objective genetic algorithm approach to optimal allocation of multi-type FACTS devices for power systems security, in: Proceedings of the IEEE Power Engineering Society General Meeting, 2006, pp. 8–16. [12] K. Deb, A. Pratap, S. Agrawal, T. Meyarivan, A fast and elitist multi-objective genetic algorithm: NSGA-II, IEEE Transactions on Evolutionary Computation 6 (2) (2002) 182–197. [13] E. Zitzler, M. Laumanns, L. Thiele, SPEA2: improving the strength pareto evolutionary algorithm, in: Proceedings of Evolutionary Methods for Design, Optimization and Control with Applications to Industrial Problems – EUROGEN, 2001, pp. 95–100. [14] E. Zitzler, S. Kunzli, Indicator-based selection in multi-objective search, in: Proceedings of the Eighth International Conference on Parallel Problem Solving from Nature – PPSN VIII, 2004, pp. 832–842. [15] F. Xue, A.C. Sanderson, R.J. Graves, Pareto-based multi-objective differential evolution, in: Proceedings of the 2003 Congress on Evolutionary Computation – CEC’2003, vol. 2, Canberra, Australia, IEEE Press, 2003, pp. 862–869. [16] P.D. Justesen, R.K. Ursem, Many-objective distinct candidates optimization using differential evolution on centrifugal pump design problems, in: Proceedings of Evolutionary Computation (CEC), IEEE Congress, 2010, pp. 1–8. [17] L.J. Eshelman, J.D. Schaffer, in: D. Whitley (Ed.), Real-Coded Genetic Algorithms and Interval Schematam, 1993, pp. 187–202. [18] J.A. Wood, F.B. Woolenberg, Power Generation, Operation and Control, 2nd edition, Wiley India Pvt Ltd., New Delhi, 2006. [19] D.P. Kothari, J.S. Dhillon, Power System Optimization, 2nd edition, Prentice Hall of India Private Ltd, New Delhi, 2011, pp. 422–423. [20] IEEE 118-Bus System, Available at: www.ee.washintgton.edu, 1996.
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[21] E.-S. E.-S. El-Araby, N. Yorino, H. Sasaki, A two level hybrid GA/SLP for FACTS allocation problem considering voltage security, Electric Power and Energy Systems 25 (2003) 327–335. [22] D.E. Goldberg, Genetic Algorithms for Search, Optimization, and Machine Learning, Addison–Wesley, Reading, MA, 1989. [23] C.M. Fonseca, P.J. Fleming, An overview of evolutionary algorithms in multiobjective optimization, Evolutionary Computation 3 (1) (1995) 1–16. [24] D. Devaraj, Improved genetic algorithm for multi-objective reactive power dispatch problem, European Transactions on Electrical power 17 (6) (2007) 569–581. [25] P. Somasundaram, K. Kuppusamy, K. Devi, Evolutionary programming based security constrained optimal power flow, Electric Power System Research 72 (2004) 137–145. [26] C. Thitithamrongchai, B. EuaArpon, Security control optimal power flow: a parallel self-adaptive differential evolution approach, Electric Power Components and Systems 23 (10) (2005) 280–298. Dr. R. Narmatha Banu is an associate professor in Department of Electrical and Electronics Engineering, Kalasalingam University, Tamil Nadu, South India. She completed her B.E. (EEE) degree in Mohammed Sathak Engineering College, Kilakarai, Tamil Nadu, India in the year 1999 and M.E. (Power System Engg) degree in Annamalai University, Chidambaram, Tamil Nadu in the year 2002. She pursued Ph.D. in the Department of Electrical Engineering, Anna University, Chennai in the year 2010. Her area of interest is Power system Security, Genetic Algorithm and FACTS devices. She has got the Young Scientist Award for the year 2009 from Tamil Nadu State Council for Science and Technology (established by Government of Tamil Nadu).
Dr. D. Devaraj is a graduate from Thiagarajar College of Engineering in Electrical and Electronics Engineering (1992). He did his Masters in Power System Engineering from Madurai Kamaraj University, Madurai (1994). He obtained his Ph.D. from the Indian Institute of Technology, Chennai (2000) with specialization in power systems engineering. His research interests include power system engineering, power system automation, power system simulation, computational intelligent techniques, intelligent control techniques. He is currently working as a senior professor and Dean (research and development) in the Kalasalingam University, Krishnankoil, Tamil Nadu.
