Genetic-algorithm-based optimal power flow for security enhancement D. Devaraj and B. Yegnanarayana Abstract: Power-system-security enhancement deals with the task of taking remedial action against possible network overloads in the system following the occurrence of contingencies. Line overload can be removed by means of generation redispatching and by adjustment of phase-shifting transformers. The paper presents a genetic-algorithm (GA) based OPF algorithm for identifying the optimal values of generator active-power output and the angle of the phase-shifting transformer. The locations of phase shifters are selected based on sensitivity analysis. To overcome the shortcomings associated with the representation of real and integer variables using the binary string in the GA population, the control variables are represented in their natural form. Also, crossover and mutation operators which can deal directly with integers and floating-point numbers are used. Simulation results on IEEE 30-bus and IEEE 118-bus test systems are presented and compared with the results of other approaches.
List of symbols Nb NPQ Ng S Pi, Qi Vi, Vj Gij, Bij Pij Pgi Qgi
1
total number of busbars number of load busbars number of generators slack bus real and reactive powers injected into network at bus i voltage magnitude at busbars i and j self conductance and susceptance of busbars i and j voltage-angle difference between bus i and j real power generation at bus i reactive power at bus i
Introduction
With the continued increase in demand for electrical energy with little addition to transmission capacity, security assessment and control have become important issues in power-system operation. Security assessment deals with determining whether or not the system operating in a normal state can withstand contingencies (such as outage of transmission lines, generators etc.) without any limit violation. If the present operating state is found to be insecure, action must be taken to prevent limit violation in the contingency state. r IEE, 2005 IEE Proceedings online no. 20045234 doi:10.1049/ip-gtd:20045234 Paper first received 3rd November 2004 and in final revised form 17th May 2005 D. Devaraj is with Department of Electrical and Electronics Engineering, AKCE, Krishnankoil, India B. Yegnanarayana is with Department of Computer Science Engineering, IIT Madras, Chennai, India E-mail:
[email protected] IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 6, November 2005
Transmission-line overload can be alleviated by rerouting power flows in the system. A change in line flow can be caused by an appropriate change in phase angles and magnitude of bus voltages, which are usually referred to as state variables. The state variables can, in turn, be modified by a variation in generated power. In [1], the linearised relationship between power flow in the overloaded lines and the generated power has been used to reschedule the power generation. A computationally simple algorithm has been developed in [2] for real-time security control. In [3], a fuzzyset-theory-based approach has been proposed for overload alleviation through real power-generation rescheduling. A generation-shift sensitivity factor was used to determine the change in generation required at a generator busbar. Although these approaches are fast in rescheduling the power generation, it may lead to overloading in other lines. For secure operation of the system without any limit violation, complete modelling of the system through loadflow equations and operational constraints is necessary [4]. Apart from generation rescheduling, adjustment of phase angles of-phase shifting transformers will also result in a change in line-flow pattern. This paper presents an optimal power flow with phase shifter for overload alleviation. The locations of phase shifters are identified based on sensitivity analysis. The OPF solution gives the optimal settings of all controllable variables for a static power-system-loading condition. A number of mathematical-programming-based techniques have been proposed to solve the OPF problem. These include the gradient method [4–6]. Newton method and linear programming [7, 8]. In [9, 10] mixed-integer linear programming has been applied to identify the location of the phase shifter and FACTS devices in order to improve the loadability of the system. The gradient and Newton methods suffer from the difficulty in handling inequality constraints. To apply linear programming, the input–output function is to be expressed as a set of linear functions, which may lead to loss of accuracy. Also, difficulties are encountered in incorporating directly the discrete variables related to the phase-shifting transformers. 899
In [11], a rule-based OPF with phase shifter has been proposed to alleviate the line overload. The principal shortcoming of a rule-base approach is that the construction of rules requires extensive help from skilled knowledge engineers. Also, it does not provide a continuous fabric over the solution space. Recently, global-optimisation techniques such as the genetic algorithm have been proposed to solve the optimalpower-flow problem [12, 14]. A genetic algorithm [15] is a stochastic search technique based on the mechanics of natural genetics and natural selection. It works by evolving a population of solutions towards the global optimum through the use of genetic operators: selection, crossover and mutation. The traditional binary-coded genetic algorithm has number of difficulties in dealing with continuous search spaces [13]. In this paper, the decision variables are represented in their natural form. Also, crossover and mutation operators which can operate directly with integer and floating-point numbers are presented. The proposed approach is illustrated through a corrective action plan for a few harmful contingencies in the IEEE 30-bus and IEEE 118-bus systems. 2
3 Mathematical formulation of optimal-powerflow problem The conventional formulation of the optimal-power-flow (OPF) problem determines the optimal settings of control variables such as real power generations, generator terminal voltages, transformer tap settings and phase-shifter angles while minimising an objective function such as fuel cost given in (3). FT ¼ Sðai Pgi2 þ bi Pgi þ ci Þ
During security control, the prime task of the power-system operator would be to remove the line overload. Hence, the minimum severity index is taken as the objective function in this paper. The minimisation problem is subjected to the constraints (i) Load-flow constraints: Pi ¼ Vi
0
@IS IS ð0Þ IS ðfÞ ¼ jDfi j @fi
Qi ¼ Vi
Nb X
Vj ðGij cos yij Bij cos yij Þ
j¼1
ð5Þ
i ¼ 1; 2; . . . ; Npq (ii) voltage constraints: Vimin Vi Vimax
i 2 Nb
ð6Þ
Pgimin Pgi Pgimax
i 2 Ng
ð7Þ
max Qmin gi Qgi Qgi
i 2 Ng
ð8Þ
(iii) Unit constraints:
fmin fi fmax i i
i 2 Nt
ð2Þ
where IS (0) ¼ total line overload before phase shifter i is installed IS(f) ¼ total line overload after phase shifter i is installed Nt ¼ set of phase-shifter transformers. In (2), SSIf40, if line overload is reduced by use of a phase shifter, i.e., IS(f)oIS(0). Obviously, if phase shifter i is not helpful in alleviating line overload, then IS(f)ZIS(0). In this case, we define the value of sensitivity SSIf ¼ 0. The locations which have high values of sensitivity are selected for placing the phase shifter. 900
ð4Þ
(iv) Phase-shifting-transformer constraint:
The line flows in (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 effects. For IEEE 30-bus and IEEE 118-bus systems considered in this work, we have fixed the value of m as 1. To determine the best location for installing a phase shifter, a sensitivity analysis is conducted. The formulation of sensitivity analysis of severity index with respect to phaseshifter variables can be expressed as SSIf ¼
Vj ðGij cos yij þ Bij sin yij Þ
i ¼ 1; 2; . . . ; Nb ; i ¼ 6 s
where ISl ¼ flow in line l (MVA) ¼ rating of the line l (MVA) Smax l L0 ¼ set of overloaded lines m ¼ integer exponent
Nb X j¼1
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: n X Sl 2m ð1Þ severity index ISl ¼ Slmax l2L
ð3Þ
i 2 Nt
ð9Þ
The power-flow equations are used as equality constraints and the inequality constraints are the limit on active and reactive power generations, phase-shifting-transformer setting, busbar voltage magnitudes and apparent power flows in branches. 4
Overview of genetic algorithm
A genetic algorithm (GA) is a general-purpose optimisation algorithm based on the mechanics of natural selection and genetics. Unlike traditional hill–climbing methods involving iterative changes to a single solution, genetic algorithms work with a population of solutions. A fitness value, derived from the problem’s objective function is assigned to each member of the population. Individuals that represent better solutions are awarded higher fitness values, thus enabling them to survive more generations. Starting with an initial random population, successive generations of populations are created by the genetic operators reproduction, crossover and mutation to yield better solutions which approach the optimal solution to the problem. The characteristics of good candidates have more chances to be inherited, because good candidates live longer. The average strength of the population therefore rises through the generations. Finally, the population stabilises, because no better individual can be found. At that stage, the algorithm has converged, and most of the individuals in the IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 6, November 2005
population are almost identical, and represent the optimal solution to the problem. A genetic algorithm is governed by three parameters: the population size, crossover rate and mutation rate. The genetic algorithm offers a promising alternative in solving combinatorial-optimisation problems. A GA does not require any space limitations such as smoothness, convexity or unimodality of the function to be optimised. This feature makes it suitable for many real-world applications including the optimal-power-flow problem. In the traditional binary-coded genetic algorithm, the decision variables of the problem are represented by a fixed-length string of binary bits (0, 1). In this representation, the resolution of the solution depends on the number of bits used to represent the variables. Further, the coding of realvalued variables in finite-length strings causes a number of difficulties. To overcome these difficulties, in this paper, the decision variables are represented in their natural form. Also, crossover and mutation operators which can operate directly with integer and floating-point genetic algorithm are presented in this Section.
4.1
Crossover operation
The crossover operator is mainly responsible for the globalsearch property of the GA. Crossover basically combines substructures of two parent chromosomes to produce new features, with the selected probability typically in the range of 0.6–0.9. In this paper, the BLX [16] crossover is applied to the floating-point numbers and a single-point crossover operator is applied to the integer variables. The details of the BLX crossover are given here. Figure 1 illustrates the BLX crossover operation for the one-dimensional case. BLX [14] uniformly picks new
I
u min
Fig. 1
where e1 ¼ u1 a ðu2 u1 Þ
ð10Þ
e2 ¼ u2 þ a ðu2 u1 Þ
ð11Þ
r : uniform random number 2 ½0; 1: Note that e1 and e2 will lie between umin and umax, the variable’s lower and upper bounds, respectively. In a number of test problems, it was observed that a ¼ 0.5 provides good results. One interesting feature of this type of crossover operator is that the point created depends on the location of both parents. If both parents are close to each other, the new point will also be close to the parents. On the other hand, if parents are far from each other, the search is more like a random search.
Reproduction
The selection of individuals to produce successive generations plays an important role in GA. Reproduction comprises forming a new population, usually with the same total number of chromosomes, by selecting from members of the current population following a particular scheme. The higher the fitness, the more likely it is that the chromosome will be selected for the next generation. There are several strategies for selecting the individuals, e.g. roulette-wheel selection, ranking methods and tournament selection. Here we use tournament selection. In tournament selection, n individuals are selected at random from the population and the best of the n is inserted into the new population for further genetic processing. This procedure is repeated until the mating pool is filled. Tournaments are often held between pairs of individuals (n ¼ 2), although larger tournaments can be held. Though this scheme selects good individuals for the next generation, it cannot guarantee that the best solution obtained survives throughout the optimisation process. In other words, the best solution obtained from the GA may not be included in the final solutions, which are clustered towards a solution. To avoid this, the GA is modified such that, once an individual with highest fitness among the current generation is found, it will be kept unchanged into the next generation. This process is called ‘elitism’.
4.2
individuals with values that lie in [I I, I+ I] where u1 and u2 represent two parents from a particular variable. The offspring y is sampled from the space [e1, e2] as follows: e1 þ r ðe2 e1 Þ : if umin y umax y¼ repeat sampling : otherwise
e1
u1
4.3
Mutation
The mutation operator is used to inject new genetic material into the population. Mutation randomly alters a variable with a small probability. A ‘uniform mutation’ operator is applied to the real variables. It randomly selects one of the variables ui, from a parent and sets it equal to a uniform random number between the variables’ lower (umin) and upper (umax) bounds. 5
Genetic-algorithm implementation
When applying GAs to solve a particular optimization problem, two main issues must be addressed: (a) representation of the decision variables; and (b) formation of the fitness function. These issues are explained in this Section.
5.1
Problem representation
In solving the OPF problem for security-control applications, two types of variable need to be determined by the optimisation algorithm: generator active-power generation Pgi and generator terminal voltages Vgi which are continuous variables, and the phase angle of the phaseshifting transformer fi, which is a discrete variable. Conventional methods are not efficient in handling problems with discrete variables. Each individual in the population represents a candidate OPF solution. We consider the generator active power, generator terminal voltage and phase angle of the phase-shifting transformer as control variables of the OPF problem. These variables are represented in their natural form, i.e generator active power Pgi and generator terminal voltages Vgi are represented as real numbers, whereas the phase angle of the phase-shifting transformer, being a discrete quantity, is represented as integer. With this representation, a typical chromosome of the OPF problem looks like the following: 97:5 100:8 Pg3 Pg2
u2
e2
Schematic representation of BLX–a crossover
IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 6, November 2005
u max
. . . 250:70 Pgn
0:975 Vg1
1:020 Vg2
...
0:90 Vgn
1 1
3
... 2
1 n
The use of floating-point numbers in the GA representation has a number of advantages over binary encoding. The 901
efficiency of the GA is increased as there is no need to convert the solution variables to the binary type. Moreover, less memory is required and there is no loss in precision by discretisation to binary or other values. Also, there is greater freedom to use different genetic operators.
5.2
Fitness function
GA searches for the optimal solution by maximising 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 OPF problem under consideration, the objective is to minimise the severity in the postcontingency state satisfying the equality and inequality constraints (4)–(9). The limits on power generations at the generator buses, voltage magnitude at the generator buses and the phase angle of the phase-shifting transformer are implicitly satisfied by the problem representation itself. The equality constraints (4) and (5) are satisfied by solving these equations using the Newton–Raphson load-flow algorithm. However, the computed real power at the slack busbar, reactive power at generator busbars, the voltage magnitude at load busbars and the power flows in all transmission lines are not guaranteed to satisfy the operating limits. These constraints on state variables are handled through a penalty-function approach. With the inclusion of a penalty function the new objective function becomes Min f ¼ IS þ PS þ
Npq X
PVi þ
i¼1
Ng X
PQi
ð12Þ
i¼1
Here PS, PVi and PQi are the penalty terms for the reference-busbar generator active-power-limit violation, load-busbar voltage-limit violation and reactive-powergeneration limit violation, respectively. These quantities are defined by the equations 8 2 < Ks ðPs Psmax Þ if Ps 4Psmax 2 min ð13Þ Ps ¼ Ks ðPs Ps Þ if Ps oPsmin : 0 otherwise 8 2 < Kv ðVj Vjmax Þ 2 VPj ¼ Kv ðVj V min Þ j : 0
if Vj 4Vjmax if Vj oVjmin otherwise
8 2 < Kq ðQj Qmax j Þ QPj ¼ Kq ðQj Qmin Þ2 j : 0
if Qj 4Qmax j if Qj oQmin j otherwise
ð14Þ
ð15Þ
where Ks, Kv and Kq are the penalty factors. The success of the approach lies in the proper choice of these penalty parameters. Using the above penalty-function approach, one has to experiment to find a correct combination of penalty parameters Ks, Kv and Kq. However, to reduce the number of penalty parameters, the constraints are often normalised and only one penalty factor R is used. During the GA run, GA searches for a solution with maximum fitness-function value. Hence, the minimisation objective function given by (3) is transformed to a fitness function to be maximised as Fitness ¼
k 1þf
where k is a constant. In the denominator a value of 1 is added with f in order to avoid division by zero in case of complete overload alleviation. 902
6
Simulation results
This Section presents the details of the study carried out on IEEE test systems for security enhancement. Three different cases were considered for the study. In the first case, the proposed GA-based algorithm was applied to obtain the optimal-control variables in the IEEE 30-bus system under normal conditions. In the second case, the proposed algorithm was applied to alleviate overloads under line outage through generator rescheduling and phase-shifting transformers. In the third case, the GA-based algorithm was applied to alleviate line overload in the IEEE 118-bus system. The GA code was written in Matlab and executed on a PC with a Pentium IV processor.
6.1 Case 1: optimal scheduling for the base case IEEE 30-bus system has six generators and 41 transmission lines. The generator and transmission-line data relevant to the system are taken from [5]. The upper and lower voltage limits at all the busbars except slack were taken as 1.10 and 0.95, respectively. The slack busbar voltage was fixed to its specified value of 1.06 p.u. Here the contingencies are not considered and the GA-based algorithm was applied to find the optimal scheduling of the power system for the basecase loading condition given in [5]. The objective function in this case is minimisation of the total fuel cost. Generator active-power outputs and the generator-busbar terminal voltages were taken as the optimisation variables. The optimisation 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 a GA for various crossover and mutation probabilities in the ranges 0.6–1.0 and 0.001–0.1, respectively, was therefore evaluated. The best results of the GA were obtained with the following control parameters: Number of generations ¼ 50 Population size ¼ 30 Crossover probability ¼ 0:8 Mutation probability ¼ 0:01 The GA took 26 s to complete the 50 generations. After 50 generations it was found that all the individuals have reached almost the same fitness value. This shows that GA has reached the optimal solution. Figure 2 shows the variation of the fitness during the GA run for the best case. It can be seen that the fitness value increases rapidly in the first 10 generations of the GA. During this stage, the GA concentrates mainly on finding feasible solutions to the problem. Then the value increases slowly, and settles down near the optimum value with most of the individuals in the population reaching that point. The optimal values of control variables along with the real-power generation of the slack busbar generator and the reactive-power generation of all the generating units are given in Table 1. The minimum cost obtained with the proposed algorithm is near to the minimum cost of $ 802.43/h, reported in [5] using the gradient method. Also, it was found that all the state variables satisfy the lower and upper limits. IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 6, November 2005
14 12
fitness
10 8 6 4 best average
2 0
0
5
10
15
20
25
30
35
40
45
50
generations
Fig. 2
Convergence of the GA-OPF algorithm
Table 1: Base-case solution of IEEE 30-bus system Busbar
Generated power
Busbar voltage magnitude (p.u.)
Real (MW)
Reactive (MV)
1
179.39
11.33
2
48.83
9.90
1.0317
5
21.84
23.73
1.0052
8
21.75
41.13
1.0125
11
12.05
31.14
1.0495
13
12.36
40.19
1.0426
conducted under base-load conditions to identify the harmful contingencies. From the contingency analysis, it was found that line outages 1–2, 1–3, 3–4 and 2–5 have resulted in overload on other lines. The power flow on the overloaded lines and the calculated value of severity index for each contingency are given in Table 2. From this Table it is found that line outage 1–2 is the most severe one, and results in overloading on three other lines. Sensitivity analysis was carried out to identify the suitable locations of the phase shifter to alleviate the line overload. The four locations identified for each contingency are given in Table 3. The GA-based OPF algorithm was applied to alleviate the line overload in all four severe-contingency cases. To test the ability of the proposed algorithm to alleviate overload under severe conditions the real and reactive load in all the load busbars were increased to 1.2 times the baseload condition. The maximum reactive-power generation of all the generators was also increased correspondingly. Generator active power and the phase-shifting transformer are taken as the control variable. The limits of the phase shifters were taken as 7100. The minimum severity index was taken as the objective function of the GA. The algorithm was run for a maximum of 25 generations and was made to stop if the targeted value of IS ¼ 0 was reached. Table 4 presents the optimal control-variable
Table 2: Summary of contingency analysis for IEEE 30-bus system
1.0443
Outage line
Overloaded lines
1–2
1–3 3–4
Fuel cost: 803.05 $/h
For comparison, the OPF problem was solved using a binary-coded genetic algorithm with 10 bits for real power generation and six bits for voltage magnitude. The genetic population was stored in a 30 86 array, compared with a 30 11 array used by the real-coded genetic algorithm. The binary-coded GA took approximately 35 s to reach the same production cost reached by the real-coded GA. This shows that the proposed algorithm occupies less computer space and take less time to reach the optimal solution.
Line flow (MVA)
Line-flow limit (MVA)
Severity index
191.24
130
5.6228
182.13
130
4–6
110.07
90
1–3
1–2
182.44
130
3.0478
3–4
1–2
179.64
130
2.9584
2–6
66.56
65
2–5
2–6
77.36
65
5–7
101.43
70
2.7252
Table 3: Phase-shifter locations in IEEE 30-bus system Line outage Phase-shifter location
1–2
1–3
3–4
2–5
2–4
6–7
6–7
1–2
6.2 Case 2: overload alleviation through generation rescheduling
2–5
2–5
2–5
2–6
In this case, the GA-based algorithm is used for corrective control under a contingency state. Contingency analysis was
2–6
5–7
2–6
2–4
4–6
4–12
5–7
4–6
Table 4: Control variable setting for corrective action in IEEE 30-bus system Line outage P1
P2
P5
P8
P11
P13
1
2
3
4
SI
Time taken (s)
1–2
128.35
77.53
48.28
32.95
27.45
39.15
2
10
5
2
0
8.42
1–3
128.95
75.20
49.50
34.68
23.69
39.60
10
10
10
1
0
11.28
3–4
131.07
74.57
49.65
30.65
27.60
38.85
10
4
5
1
0
8.28
2–5
146.17
59.20
49.68
33.72
28.52
37.12
0
0
3
4
1.0293
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12.77 903
settings for all the four cases, along with the final value of severity-index. The time taken in each case is given in the last column. From this Table it is evident that the overloading of the transmission lines has been completely alleviated in a reasonable amount of time in all the cases except for line outage 2–5, in which line flow through line 5–7 was 70.52 MVA against its rating of 70 MVA. This shows the effectiveness of the proposed algorithm for overload alleviation.
Table 7: Control-variable setting for corrective action in IEEE 118-bus system Line outage 8–5
Line outage 64–65
P10
413.8
P80
579.3
P10
204.84
P80
P12
248.3
P87
206.9
P12
127.73
P87
94.33
P25
289.7
P89
620.7
P25
109.78
P89
497.56
P26
206.9
P100
320.7
P26
247.87
P100
279.55
P31
620.7
P103
82.8
P31
82.85
P103
95.56
6.3 Case 3: overload alleviation in IEEE 118bus system
P46
206.9
P111
206.9
P46
66.11
P111
19.66
P49
434.5
1
2
P49
283.87
1
2
In this case, the proposed algorithm was applied to alleviate the line overload under contingency condition in the IEEE 118-bus system. The test system has 54 generator buses and 186 transmission lines. The real-power generation Pg of the generators is given in Appendix 1 (Section 9.1) and the line rating of the transmission line is given in Appendix 2 (Section 9.2). All other data are the same as the standard IEEE 118-bus data [17]. Contingency analysis was conducted on this system and the top two severe-contingency cases are produced in Table 5 along with the overloaded lines and the severity index. Based on the sensitivity analysis, six locations were identified for the installation of a phase-shifting transformer and the GA-based algorithm was applied to reschedule the generator output and to identify the parameters of the phase-shifting transformer. The optimal setting of the phase-shifting-transformer position and generator output to alleviate the overload are given in Tables 6 and 7.
P54
620.7
2
2
P54
89.21
2
2
P59
372.4
3
7
P59
219.37
3
3
P61
620.7
4
4
P61
202.34
4
6
P65
413.8
5
0
P65
214.32
5
9
P66
413.8
6
7
P66
438.90
6
7
P69
620.7
P69
799.15
7
Conclusions
Table 5: Summary of severe contingency cases for IEEE 118-bus system Outage
Overloaded
Line flow
Line-flow
Severity
line
lines
(MVA)
limit (MVA)
index
8–5
12–14
107.20
100
6.2002
13–15
103.07
100
12–16
143.65
130
15–17
223.48
200
16–17
160.22
130
89–92
206.60
200
100–103
123.75
100
9–10
445.9
130
65–66
172.79
100
66–67
108.90
100
64–65
Table 6: Phase-shifter locations Line outage
8–5
64–65
Phase-shifter location
3–5
56–57
904
This paper has proposed a flexible optimisation tool based on a genetic algorithm to aid power-system operators. The application of this tool for scheduling the power system during the normal operation and to schedule the powersystem controls during contingencies has been presented. The line overloads were relieved through rescheduling of generator outputs and adjustment of the phase angle of a phase-shifting transformer. To alleviate the line overloads effectively, the phase-shifter location is identified based on sensitivity analysis. In this paper, the problem of discretisation in the representation of the decision variables in the binary-coded GA has been alleviated by employing floating-point numbers to represent the generator loadings and integers for the phase angles. A modified form of crossover and mutation operations to deal with the real and integer variables has been presented. Compared with a conventional GA, the proposed algorithm occupies less computer space and takes less CPU time and is well suitable for real-time applications. 8
4.1716
3–12
38–65
17–18
62–67
18–19
65–66
30–17
66–67
39–40
65–68
313.09
References
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11 12 13
14 15 16 17
using MILP’. Proc. Int. Conf. Energy, Information Technology and Power Sector, Kolkata, Jan. 2005, pp. 515–523 Momoh, J.A., Zhu, J.Z., Boswell, G.D., and Hoffman, S.: ‘Power system security enhancement by OPF with phase shifter’, IEEE Trans., 2001, PWRS-16, (2), pp. 287–293 Goldberg, D.: ‘Genetic algorithms in search, optimization and machine learning’ (Addison–Wesley, 1989) Lai, L.L., Ma, J.T., Yokayama, R., and Zhao, M.: ‘ Improved genetic algorithms for optimal power flow under both normal and contingent operation states’, Int. J. Electr. Power Energy Syst., 1997, 9, (5), pp. 287–292 Paranjothi, S.R., and Anburaja, K.: ‘Optimal power flow using refined genetic algorithms’, Electr. Power Compon. Syst., 2002, 30, pp. 1055–1063 Devaraj, D., and Yegnanarayana, B.: ‘A combined genetic algorithm approach for optimal power flow’. Proc. 11th National Power Systems Conf., Bangalore, India, 2000, Vol. 2, pp. 524–528 Eshelman, L.J., and Schaffer, J.D.: ‘Real-coded genetic algorithms and interval schemata’ (D. Whitley, 1993), pp. 187–202 IEEE 118-bus system (1996), (Online) Available at //www.ee. washington.edu
9 Appendix Table 8: Generator data Bus
Pg (MW)
Bus
10
450.0
54
12
85.0
25 26
Pg (MW)
Bus
Pg (MW)
48.0
87
4.0
59
155.0
89
607.0
220.0
61
160.0
100
252.0
314.0
65
391.0
103
40.0
31
7.0
66
392.0
111
36.0
46
19.0
69
516.0
49
104.0
80
477.0
IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 6, November 2005
Table 9: Transmission-line data Transmission lines
Rating (MW)
8–9, 8–5, 9–10
650
23–24, 26–25, 30–17, 8–30, 26–30, 38–37, 30–38, 38–65, 64–65, 65–68, 68–69
600
18–19, 23–32, 64–61, 70–74, 68–81,
300
19–20, 15–19, 20–21, 21–22, 22–23, 23–25, 15 –33, 19–34, 33–37, 34–37, 60–61, 63–59, 49–66, 49–66, 69–70, 24–70, 70–71, 24–72, 70–75, 69–75, 74–75, 69–77, 75–77, 77–80, 81–80, 89–92, 89–92, 100–103, 68–116
25–27, 63–64, 71–72, 77–80,
4–5, 15–17, 88–89, 89–90
200
3–5, 5–6, 4–11, 5–11, 14–15, 12–16, 16–17, 17–18, 17–31, 27–32, 37–39, 37–40, 39–40, 40–41, 43–44, 34–43, 44–45, 47–49, 42–49, 42–49, 45–49, 48–49, 49–50, 49–51, 59–61, 47–69, 49–69, 76–77, 78–79, 79–80, 82–83, 83–85, 85–86, 86–87, 85–88, 85–89, 89–90, 90–91, 92–93, 92–94, 80–96, 80–97, 80–98, 80–99, 92–100, 94–100, 100–104, 103–104, 103–105, 100–106, 104–105, 103–110, 110–112, 17–113, 32–113, 75–118, 76–118
130
1–2, 1–3, 6–7, 11–12, 2–12, 3–12, 7–12, 11–13, 12–14, 13–15, 27–28, 28–29, 29–31, 31–32, 35–36, 35–37, 34–36, 40–42, 41–42, 45–46, 46–47, 46–48, 51–52, 52–53, 53–54, 49–54, 49–54, 54–55, 54–56, 55–56, 56–57, 50–57, 56–58.51–58.54–59.56–59.56–59, 55–59, 59–60, 60–62, 61–62, 62–66, 62–67, 65–66, 66–67, 71–73, 77–78, 77–82, 83–84, 84–85, 91–92, 93–94, 94–95, 82–96, 94–96, 95–96, 96–97, 98–100, 99–100, 100–101, 92–102, 101–102, 105–106, 105–107, 105–108, 106–107, 108–109, 109–110, 110–111, 32–114, 27–115, 114–115, 12–117
100
905
