Power System Security Improvement Computer Science Essay

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Optimal power flow is an optimizing tool for operation and planning of modern power systems. This OPF problem involves the optimization of various types of objective functions while satisfying a set of operational and physical constraints while keeping the power outputs of generators, bus voltages, shunt capacitors/reactors and transformers tap settings in their limits. This chapter describes efficient and reliable evolutionary programming (EP) and particle swarm optimization (PSO) algorithms for solving the optimal power flow problem for security enhancement. The objectives considered are: cost of generation, severity index based line loadings and severity index based on fuzzy logic composite criteria. The results are obtained on IEEE 30-bus system and have been used to show the effectiveness of the proposed algorithms. Comparisons were made between the approaches in terms of the solution quality and convergence characteristics.

1. INTRODUCTION

The problem of power system security has obtained much attention in the operation of present day power systems. To meet the load demand in a power system and satisfy the stability and reliability criteria, either the existing transmission lines must be utilized more efficiently, or new line(s) should be added to the system. The latter is often impractical. The reason is that building a new power line is a very time consuming process and sometimes an impossible task, due to environmental problems.

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.

Contingency screening and ranking is one of the important components of on-line system security assessment of the modern power systems. The contingency ranking methods those reported in the literature, generally, ranks the contingencies in an approximate order of severity with respect to a scalar performance index (PI), which quantifies the system stress [1]. The common disadvantages of several PI-based contingency ranking methods are the masking phenomenon.

Moreover, with increased loading of existing power transmission systems, the problem of voltage stability and voltage collapse, has also become a major concern in power system planning and operation. It has been observed that voltage magnitudes do not give a good indicator of proximity to a voltage stability limit and voltage collapse [2, 3]. Therefore, it is necessary to consider voltage stability indices as pre/post-contingency quantities in the evaluation of severity of network contingency.

To measure the severity level of voltage stability problems, a lot of performance indices have been proposed [4]. They could be used on-line or off-line to help operators determine how close the system is to collapse. In general, these indices aim at defining a scalar magnitude that can be monitored as system parameters change, with fast computation speed. They include the sensitivity factors [5], second order performance index [6], voltage instability proximity index (VIPI) [7], singular values and eigen values [8], and so on.

This paper presents a new approach for assessment of the power system security. Using fuzzy membership functions of post-contingent quantities, it quantifies the security state of a power system, which uses off-line screening for the most vulnerable system states. It introduces Network Composite Overall Severity Index (NCOSI) of the power system using system's variables characterized by fuzzy sets of a trapezoidal form [9]. The NCOSI uses the voltage stability indices at the load buses as post-contingent quantities in addition to real power loadings and bus voltage violations to evaluate the network contingency.

For secure operation of the system without any limit violation, complete modeling of the system through load flow equations and operational constraints is necessary. Thus the Optimal Power Flow (OPF) is a good choice. The solution of formulated optimal power flow model gives the optimal operating state of a power system and the corresponding settings of control variables for economic and secure operation, while at the same time satisfying various equality and inequality constraints. The equality constraints are the power flow equations, while the inequality constraints are the limits on control variables and the operating limits of power system dependent variables. Amongst a number of different operational objectives that an OPF problem may be formulated, a widely considered objective is to minimize the fuel cost subject to equality and inequality constraints.

The goal of optimal power flow is to determine optimal control variables and quantities for efficient power system planning and operation. Several optimization techniques have been proposed to handle the OPF problem [10-12]. A wide variety of optimization techniques have been applied to solve the OPF problems such as nonlinear programming (NLP) [10,13-17], quadratic programming [18,19], linear programming [20,21], Newton-based techniques [22,23], sequential unconstrained minimization technique [24]. Generally NLP based procedures have many drawbacks such as insecure convergence properties and algorithmic complexity.

Quadratic programming based techniques have some disadvantages which are associated with piecewise quadratic cost approximation. Newton-based techniques have the drawback of the convergence characteristics that are sensitive to the initial conditions and they may even fail to converge due to the inappropriate initial conditions. Sequential unconstrained minimization techniques are known to exhibit numerical difficulties when the penalty factors become extremely large. Although linear programming methods are fast and reliable they have some disadvantages associated with the piecewise linear cost approximation.

In the recent past, the research in OPF such as interior point method (IPM) has been gaining wider attention in power system operation [25, 26]. The interior point method is faster and more reliable for achieving feasibility and convergence and has been reported as computationally efficient, however if step size is not chosen properly, the sub-linear problem may have a solution that is infeasible in the original non-linear domain [27].

Heuristic algorithms such as Genetic Algorithms (GA) [28] and evolutionary programming [29] have been proposed for solving the OPF problem. The results reported were promising and encouraging for further research in this direction. Unfortunately, the research has identified some deficiencies in the GA performance [30].

Recently, Evolutionary Programming (EP) based approach is quite encouraging to solve OPF problem. The EP technique is a stochastic optimization method in the area of evolutionary computation, which uses the mechanics of evolution to produce optimal solutions to a given problem. It works by evolving a population of candidate solutions towards the global minimum through the use of a mutation operator and selection scheme. The EP technique is particularly well suited to non-monotonic solution surfaces where many local minima may exist.

Another evolutionary computation technique, called Particle Swarm Optimization (PSO), has been proposed and introduced [31-34]. This technique combines social psychological principles in socio-cognition human agents and evolutionary computations. PSO has been motivated by the behavior of organisms such as fish schooling and bird flocking. Generally, PSO is characterized as simple in concept, easy to implement and computationally efficient. Unlike other heuristic techniques, PSO has a flexible and well-balanced mechanism to enhance and adapt to the global and local exploration abilities.

This paper presents the EP and PSO based algorithms for solving OPF problems for power system security enhancement under the selected network contingencies. The severity of the network contingency is evaluated using the fuzzy logic based severity index. The optimal control variables are obtained for the base case load condition before evaluating the network contingency ranking. In this paper, different objective functions are considered to enhance power system security. The proposed approach has been tested on IEEE 30-bus systems. The potential and effectiveness of the proposed approach are demonstrated.

2. VOLTAGE STABILITY INDEX (L-INDEX) COMPUTATION

Consider a system where is the total number of buses with 1,2…g generator buses, g+1,g+2,……,g+s SVC buses(g), g+s+1,…,n remaining buses(r =n-g-s), and t number of OLTC transformers. A load flow result is obtained for a given system operating condition. Using the load flow results the L-index [35] is computed as

(1)

where j=g+1,….,n and all the terms within the sigma on the RHS of equation (1) are complex quantities. The values are obtained from the Y-bus matrix as follows

(2)

where represents currents and voltages at the generator nodes and load nodes. Rearranging equation (2) we get

(3)

where are the required values.

The L-indices for a given load condition are computed for all load buses and maximum value of the L-indices gives the proximity of the system to voltage collapse. An L-index value away from 1 and close to 0 indicates an improved voltage stability margin.

3 SEVERITY INDEX BASED ON LINE LOADING

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 [36]:

Severity index (4)

where

=flow in line (MVA)

= rating of the line (MVA)

= set of transmission lines

= integer exponent

= Severity Index of line flow

The line flows are obtained from Newton-Raphson load-flow calculations. While using the above severity index for security assessment, all the transmission lines are considered. For IEEE30-bus system considered in this work, the value of m has been fixed as 1.

4 SEVERITY INDEX BASED ON FUZZY LOGIC COMPOSITE CRITERIA

Fuzzy set theory is a mathematical concept proposed by Prof. L. A. Zadeh in 1965. Fuzzy logic is a kind of logic using graded or quantified statements rather than ones that are strictly true or false. The fuzzy sets representing linguistic variables allow objects to have grade of membership from 0 to 1. The intervals for fuzzification for linguistic variables are determined based on the pre/post- contingent quantities. The input and output membership functions are selected based on the classification of pre/post contingent quantities. The pre/post-contingent quantities are first expressed in fuzzy set notation before they can be processed by the fuzzy reasoning rules.

The Network Composite Overall Severity Index (NCOSI) is obtained by using the parallel operated fuzzy inference systems, as shown in Figure 1, for the pre/post contingency operating conditions [9]. The overall severity index for line loading, voltage profiles, and voltage stability indices are added and the sum is used for contingency ranking which is designated as the NCOSI based severity index.

When the overall NCOSI severity index for each contingency in the contingency list has been figured out, the overall severity indices for those contingency cases with a severity index exceeding a pre-specified value are listed out and ranked according to the fuzzy logic composite criteria based severity index.

Figure 1 Parallel operated fuzzy inference systems

5 MATHEMATICAL MODEL OF OPF PROBLEM

The OPF problem is to optimize the steady state performance of a power system in terms of an objective function while satisfying several equality and inequality constraints. Mathematically, the OPF problem can be formulated as given

Min (5)

Subject to (6)

(7)

where x is a vector of dependent variables consisting of slack bus power , load bus voltages , generator reactive power outputs , and the transmission line loadings , Hence, x can be expressed as given

(8)

where NL,NG and nl are number of load buses, number of generators and number of transmission line respectively.

u is the vector of independent variables consisting of generator voltages VG, generator real power outputs except at the slack bus , transformer tap settings T, and shunt VAR compensations . Hence u can be expressed as given

(9)

Where NT and NC are the number of the regulating transformers and shunt compensators, respectively. F is the objective function to be minimized. g is the equality constraints that represents typical load flow equations and h is the system operating constraints

Objectives

The objectives considered for minimization are as follows.

Objective Function 1: Fuel cost of generating units ()

Objective Function 2: Severity index based on line loadings ()

Objective Function 3: Network Composite Overall Severity Index (NCOSI) ()

where

= min (10)

= min= (11)

= min (OSILL+OSIVP+OSIVSI) = FLCC (12)

m = integer exponent (fixed as 1)

Constraints

The OPF problem has two categories of constraints:

Equality Constraints: These are the sets of nonlinear power flow equations that govern the power system, i.e,

(13)

(14)

where

and are the real and reactive power outputs injected at bus- respectively, the load demand at the same bus is represented by and , and elements of the bus admittance matrix are represented by and .

Inequality Constraints: These are the set of constraints that represent the system operational and security limits like the bounds on the following:

1) generators real and reactive power outputs

(15)

(16)

2) voltage magnitudes at each bus in the network

(17)

3) transformer tap settings

(18)

4) reactive power injections due to capacitor banks

(19)

5) transmission lines loading

(20)

6) voltage stability index

(21) Handling of Constraints: There are different ways to handle constraints in evolutionary computation optimization algorithms. In this thesis, the constraints are incorporated into fitness function by means of penalty function method, which is a penalty factor multiplied with the square of the violated value of variable is added to the objective function and any infeasible solution obtained is rejected.

To handle the inequality constraints of state variables including load bus voltage magnitudes and output variables with real power generation output at slack bus, reactive power generation output, and line loading, the extended objective function can be defined as:

(22)

where

, , , are penalty constants for the real power generation at slack bus, the reactive power generation of all generator buses or PV buses and slack bus, the voltage magnitude of all load buses or PQ buses, and line or transformer loading, respectively. ,, , are the penalty function of the real power generation at slack bus, the reactive power generation of all PV buses and slack bus, the voltage magnitudes of all PQ buses, and line or transformer loading, respectively. NL is the number of PQ buses. The penalty function can be defined as:

, if

= , if

= , if (23)

where is the penalty function of variable , and are the upper limit and lower limit of variable , respectively.

SOLUTION TECHNIQUES OF OPF PROBLEM

6.1 Classical Methods

Many numerical techniques had been developed and used for solving OPF problem, such as Gradient method, Lagrange Newton method, Interior point method and others. However, these classical methods are designed for purely continuous-variable OPF.

Heuristics Methods

6.2.1 Evolutionary Programming

Evolutionary programming (EP) is a stochastic optimization technique, which places emphasis on the behavioral linkage between parents and their offspring. EP is a powerful optimization method, which does not depend on the first and second derivatives of the objective function and the constraints of the problem. The main advantage of evolutionary programming is that it uses only the objective function information and hence independent of the nature of the search space such as smoothness, convexity etc.

Evolutionary programming, which is a probabilistic search technique, generates the initial parent vectors distributed uniformly in intervals within the limits and obtains global optimum solution over number of iterations. The degree of optimality of each of the individual solutions is measured by their fitness. The individuals in each population compete with each other in competition scheme.

For the optimization to take place, the competition scheme must be such that the more optimal solutions have a greater chance of survival than the poorer solutions. Through this competition scheme the population evolves towards the global optimal point. The evolutionary programming method is iterative and the process is terminated by a stopping rule. The rule widely used is stop after a specified number of iterations.

Initialization: The initial population is generated after satisfying the limit constraints using a uniform random number distribution within its feasible range of all control variables.

Fitness of Candidate Solution: Using penalty factors, the constrained optimization problem is converted into unconstrained optimization problem. Each individual solution is assigned a fitness to measure its optimality with respect to the objective being optimized. To handle the constraints, penalty functions are included in fitness function.

Mutation: Using the mutation operator, a new population is produced from the existing population. An offspring vector is created from each parent vector by adding Gaussian random variable with zero mean and standard deviation, denoted as.

Competition and Selection: A selection mechanism is used to produce a new population from the two existing populations. To get the optimal values, the fitter or more optimal solutions should have a greater chance of selection.

Stopping Criteria: Optimization process is repeated until there is no appreciable improvement in the fitness value or maximum number of iterations reached.

Particle Swarm Optimization

Particle swarm optimization (PSO) is one of the efficient evolutionary computation techniques based on swarm intelligence. The PSO technique provides a population based search procedure in which individuals called particles change their positions with time.

Basic steps of PSO: The basic steps of PSO technique are briefly defined as follows:

Particle Position,: Within a population, each individual represents a candidate solution and it is represented by a 'd' dimensional vector. Let be the position of the particle.

Particle Velocity,: It is the velocity of the moving particles represented by a d-dimensional vector and the velocity of the particle is given by . It is bounded between the limits.

Individual Best, : When a particle moves through the search space, it compares its fitness value at the current position to the best previous fitness value. The best position of the particle that is associated with the best fitness encountered so far is called and its vector representation is given by. The fitness of the objective function for the of the particle is determined by the following relation

(24)

Global Best, : It is the best position among all individual best positions achieved so far and is given by. The global best can be determined by

(25)

Velocity Updation: Using the global best and individual best of each particle, the particle velocity in the dimension is updated according to the following equation.

(26)

The constants C1 and C2 are the acceleration constants and represent the weighting of stochastic acceleration terms that pull each particle towards and positions. k represents the current iteration and rand1 and rand2 are two random numbers in the range 0 and 1.

The cognitive part of PSO is represented by the second term of Eq. (26) where the particle changes its velocity based on its own thinking and memory. The social part of PSO is represented by the third term where the particle changes its velocity based on the social-psychological adaptation of knowledge. If a particle violates the velocity limits, its velocity will be set equal to the limit. represents the inertia weight and is a control parameter that is used to control the impact of the previous velocities on the current one.

Position Updation: Each particle changes its position according to the following equation with updated velocities:

(27)

Stopping Criteria: The optimization process will terminate if it reaches the pre-defined maximum number of iterations.

7 OVERALL COMPUTATIONAL PROCEDURE FOR SOLVING THE PROBLEM

The implementation steps of the proposed EP/PSO based algorithm can be written as follows;

Step 1: Input the system data for load flow analysis

Step 2: Run the power flow under the selected network contingency (transformer/transmission line outage)

Step 3: Evaluate the severity of the network contingency using fuzzy logic composite criteria based approach.

Step 4: Repeat steps 2 and 3 for all the transmission lines/transformers

Step 5: Select most severe network contingencies based on the fuzzy logic composite criteria based severity index.

Step 6: At the generation Gen =0; set the simulation parameters of EP/PSO parameters and randomly initialize k individuals within respective limits and save them in the archive.

Step 7: For each individual in the archive, run power flow under the selected network contingency to determine load bus voltages, angles, load bus voltage stability indices, generator reactive power outputs and calculate line power flows.

Step 8: Evaluate the penalty functions

Step 9: Evaluate the objective function values and the corresponding fitness values for each individual.

Step 10: Find the generation local best xlocal and global best xglobal and store them.

Step 11: Increase the generation counter Gen = Gen+1.

Step 12: Apply the EP/PSO operators to generate new k individuals

Step 13: For each new individual in the archive, run power flow to determine load bus voltages, angles, load bus voltage stability indices, generator reactive power outputs and calculate line power flows.

Step 14: Evaluate the penalty functions

Step 15: Evaluate the objective function values and the corresponding fitness values for each new individual.

Step 16: Apply the selection operator of EP/PSO and update the individuals.

Step 17: Update the generation local best xlocal and global best xglobal and store them.

Step 18: If one of stopping criterion have not been met, repeat steps 7-17. Else go to stop 19

Step 19: Print the results

There are two stopping criterion for the optimization algorithm. The algorithm can be stopped if the maximum number of generations is reached (Gen = Genmax) or there is no solution improvement over a specified number of generations. The first criterion is used in this chapter.

8 SIMULATION RESULTS

The proposed EP and PSO algorithms for solving optimal power flow problems are tested on standard IEEE 30-bus test system. The proposed algorithms are implemented using MATLAB 7.8 running on Core2Duo , 2.20GHz, 3GB RAM personal computer. The EP and PSO parameters used for the simulation are summarized in Table 1

Table 1 Optimal parameter settings for EP and PSO

Parameter

EP

PSO

Population size

Number of iterations

Cognitive constant, c1

Social constant, c2

Inertia weight, W

20

250

-

-

-

20

150

2

2

0.3-0.95

IEEE 30-bus system results

This section presents the details of the study carried out on IEEE-30 bus test system for security enhancement. In the IEEE 30-bus system there are 41 branches, six generator buses and 24 load buses.

Three different cases were considered for the study. In the first case, the proposed PSO based algorithm was applied to obtain the optimal-control variables in the IEEE 30-bus system under base load conditions. In the second case, the proposed fuzzy logic composite criteria were applied for network contingency ranking. In the third case, the proposed fuzzy logic composite criteria and PSO algorithm was applied to alleviate overloads under selected set of severe network contingency through the solution of optimal power flow.

Case 1: Base case condition

The proposed PSO algorithm was applied to find the optimal scheduling of the power system for the base case loading condition. The objective function in this case considered is minimization of total fuel cost. Generator active-power outputs, generator terminal voltages, transformer tap settings and shunt reactive power compensating elements were taken as control variables. The control variables are represented as floating point numbers in the population. The upper and lower voltage limits of load buses were taken as 1.06 and 0.95 respectively.

The optimal values of control variables along with the real power generation of the slack bus generator are given in Table 2. The minimum cost obtained with the PSO algorithm is $800.966/h, which is less than the minimum generation cost of $803.1916/h obtained with interior point method. Also, it was found that all the state variables satisfy the lower and upper limits. For comparison, the OPF problem was solved using an evolutionary programming method with the population size of 20 and 250 generations. All the solutions satisfy the constraints on control variable limits and line flow limits. The convergence of generation cost is shown in Figure 2. From the Figure 2, it can be observed that the PSO took approximately 60 generations to reach the same production cost reached by EP. This shows that the proposed PSO algorithm occupies less computer space and takes less time to reach the optimal solution.

Table 2 Base case solution for IEEE 30-bus system

Control Variables

(p.u.)

Base

Case

Proposed Methods

IPM

EP

PSO

Real power generation

PG1

PG2

PG3

PG4

PG5

PG6

VG1

VG2

VG3

VG4

VG5

VG6

Tap-1

Tap-2

Tap-3

Tap-4

QSVC1

QSVC2

QSVC3

QSVC4

QSVC5

QSVC6

QSVC7

QSVC8

QSVC9

0.9873

0.80

0.20

0.20

0.50

0.20

1.050

1.045

1.010

1.050

1.010

1.050

0.978

0.969

0.932

0.968

0

0

0

0

0

0

0

0

0

1.7735

0.4877

0.2150

0.1209

0.2148

0.1200

1.0700

1.0450

1.0100

1.0500

1.0100

1.0500

0.9780

0.9690

0.9320

0.9680

0

0

0

0

0

0

0

0

0

1.7642

0.4880

0.2257

0.1103

0.2130

0.1239

1.0700

1.0567

1.0335

1.0849

1.0322

1.0496

1.0077

1.0401

1.0023

0.9791

0.0738

0.0807

0.0873

0.0629

0.0609

0.0402

0.0222

0.0348

0.0434

1.7808

0.4823

0.2051

0.1236

0.2142

0.1200

1.0700

1.0538

1.0355

1.1000

1.0299

1.0595

1.0650

0.9566

1.0182

0.9942

0.0567

0.1000

0.0671

0.0214

0.0501

0.0720

0.0748

0.0541

0.0010

Generator voltages

Transformer tap

Shunt

Compensation

Cost($/h)

Ploss(p.u.)

Ljmax

CPU time(s)

900.5995

0.0533

0.1402

-

803.1916

0.0979

0.1385

0.8410

801.0204

0.0911

0.1371

580.2810

800.966

0.0920

0.1351

325.8600

Figure 2 Convergence of generation cost

The comparison of fuel cost of the proposed methods with those of the methods reported in the literature is given in Table 3. It can be seen from the Table 3 that the PSO algorithm gives less cost of generation compared with the cost of generation obtained with other methods.

Table 3 Comparison of fuel costs

Method

Fuel Cost ($/hr)

EP [37]

802.9070

TS [37]

802.5020

TS/SA [37]

802.7880

ITS [37]

804.5560

IEP [37]

802.4650

SADE_ALM [38]

802.4040

OPFPSO [39]

800.4100

MDE-OPF [40]

802.3760

Genetic Algorithm ($/hr) [36]

803.0500

Gradient method [41]

802.4300

EP (proposed)

801.0204

PSO (proposed)

800.9660

Case 2: Contingency ranking using fuzzy approach

In this case, the proposed fuzzy logic approach for network contingency ranking was applied with the optimal base case control variables under base case load conditions to identify the harmful contingencies. From the contingency analysis, it was found that line outages 2-5, 11-13 and 8-11 have resulted in heavy overload on other line and are ranked as top 3 contingencies and are given in Table 4. The severity indices of line loadings, voltage profiles and voltage stability indices for the top 10 network contingencies are evaluated and are given in Table 5. If any one of the index among OSILL, OSIVP, OSIVSI is considered then the ranking may not yield actual severity of a contingency. From Table 5, it can be observed that, consideration of all the indices is an effective procedure for evaluating the severity of a network contingency.

The number of lines and the number of load buses under each severity category for the top 10 network contingencies are given in Table 6. From Tables 5 and 6, it is found that the line outage 2-5 is the most severe one, and results in maximum total severity index compared to other lines.

Table 4 Summary of overloaded lines for IEEE-30 bus system

Line

Outage

Overloaded

Lines

Line flow

(MVA)

Line flow

Limit (MVA)

2-5

2-13

5-7

75.83

81.77

65

70

11-13

1-2

2-13

132.89

71.91

130

65

8-11

1-2

2-13

181.08

66.72

130

65

1-8

1-2

2-13

183.86

67.64

130

65

Table 5 Overall severity indices and ranking for IEEE-30 bus system.

Line

outage

OSILL

OSIVP

OSIVSI

FLCC

Rank

2-5

649.3991

1650.7

96.4105

2396.5

1

11-13

547.2608

1668.5

96.4105

2312.2

2

8-11

531.8282

1654.0

96.4105

2282.2

3

13-7

376.3138

1798.4

96.4105

2271.2

4

1-8

536.9544

1632.9

96.4105

2266.3

5

27-30

439.7578

1720.5

103.3392

2263.6

6

13-3

391.6967

1774.8

96.4105

2262.9

7

24-25

371.9588

1784.3

96.4105

2252.7

8

1-2

586.5918

1563.3

96.4105

2246.3

9

27-29

416.3521

1721.2

96.4105

2233.9

10

Table 6 Number of lines/buses under different severity category before optimization

Line outage

Line Loadings

Bus Voltage Profiles

Bus Voltage Stability Indices

R

A

N

K

LS

BS

AS

MS

BS

AS

MS

VLS

LS

BS

AS

MS

2-5

28

8

2

2

0

8

16

24

0

0

0

0

1

11-13

31

7

0

2

0

8

16

24

0

0

0

0

2

8-11

33

4

1

2

0

8

16

24

0

0

0

0

3

13-7

34

5

1

0

0

6

18

24

0

0

0

0

4

1-8

33

4

1

2

0

8

16

24

0

0

0

0

5

27-30

32

6

2

0

0

7

17

23

1

0

0

0

6

13-3

33

6

1

0

0

7

17

24

0

0

0

0

7

24-25

34

5

1

0

0

6

18

24

0

0

0

0

8

1-2

33

4

0

3

0

9

15

24

0

0

0

0

9

27-29

33

5

2

0

0

7

17

24

0

0

0

0

10

Case 3: OPF for overload alleviation

To test the ability of the proposed PSO algorithm for solving optimal power flow problem for security enhancement, it was applied under the selected three most severe network contingencies. Three objective functions are considered for the minimization using the proposed PSO algorithm.

Figure 3(a) Convergence of objective function-1 under rank -1 contingency

Figure 3(b) Convergence of objective function-2 under rank-1 contingency

Figure 3(c) Convergence of objective function-3 under rank-1 contingency

Figure 3 Convergence characteristics of objective functions for IEEE 30-bus system

Figures 3 shows the convergence characteristics of the three objective functions under the rank-1 network contingency. It can be observed that the EP converge to lower values than PSO during initial evolutions and the PSO converge to a minimum value than EP after 20 iterations. Tables 7 presents the optimal settings of the control-variables under the rank 1 network contingency with the three objective functions. From the Tables 7, it was found that all the state variables satisfy their lower and upper limits.

Table 7 Optimal settings of control variables under rank 1 contingency

Control variables

(p.u)

Base case (under contingency)

Objective functions and methods

Objective Function1

Objective Function2

Objective Function3

EP

PSO

EP

PSO

EP

PSO

PG1

PG2

PG3

PG4

PG5

PG6

1.8607

0.4823

0.2051

0.1236

0.2142

0.1200

1.6288

0.4308

0.3392

0.1783

0.2697

0.1261

1.6263 0.4395 0.3500 0.1653 0.2655 0.1278

0.7144

0.7910

0.3500

0.2991

0.5000

0.2468

0.6868 0.8000 0.3500 0.3000 0.5000 0.2634

0.9874

0.6608

0.2901

0.1881

0.5000

0.2889

1.0132

0.6666

0.2832

0.2043

0.5000

0.2516

VG1

VG2

VG3

VG4

VG5

VG6

1.0700

1.0538

1.0355

1.1000

1.0299

1.0595

1.0500

1.0412

1.0198

1.0733

0.9774

1.0773

1.0500

1.0481

1.0322

1.0662

0.9851

1.0901

1.0500

1.0430

1.0168

1.0140

0.9789

1.0224

1.0500

1.0454

1.0153

1.0186

0.9759

1.0257

1.0500

1.0373

1.0148

1.0174

0.9862

0.9891

1.0500

1.0366

1.0021

0.9770

0.9738

1.0358

Tap-1

Tap-2

Tap-3

Tap-4

1.0650

0.9566

1.0182

0.9942

0.9871

1.0453

1.0162

0.9616

1.0170

0.9873

1.0255

0.9854

1.0094

0.9920

0.9783

0.9947

0.9985 0.9945 1.0012 0.9818

1.0524

1.0196

0.9616

0.9858

1.0135

0.9523

0.9665

0.9573

Qsh10

Qsh12

Qsh15

Qsh17

Qsh20

Qsh21

Qsh23

Qsh24

Qsh29

0

0

0

0

0

0

0

0

0

0.0014

0.0082

0.0539

0.0571

0.0393

0.0914

0.0437

0.0327

0.0481

0.0278

0.0349

0.0368

0.0164

0.0254

0.0240

0.0148

0.0359

0.0278

0.0940

0

0.0364

0.0626

0.0415

0.0897

0.0136

0.1000

0.0315

0.1000 0.1000 0.0522 0.0351 0.0417 0.0799 0.0306 0.0595 0.0242

0.0185

0.0474

0.0940

0.0119

0.0847

0.0954

0.0075

0.0145

0.0546

0.0133

0.0387

0.0039

0.0229

0.0349

0.0247

0.0096

0.0396

0.0351

P-Loss

Cost

LFsi

OSILL

OSIVP

OSIVSI

FLCC

0.1719

827.8436

10.0781

649.3991

1650.7

96.4105

2396.5

0.1389

828.6224

8.3903

518

1102

96

1716

0.1404

828.5733

8.8761

573

995

96

1664

0.0673

941.5172

5.3293

386

866

96

1347

0.0662

946.1299

5.2954

385

871

96

1352

0.0813

906.4696

6.4450

360

864

96

1320

0.0850

903.5629

6.5864

360

864

96

1320

From the Tables 7, it can be observed that the PSO algorithm is able to reduce the cost of generation less than that of the cost of generation obtained by the EP method. It is also evident from the results that particle swarm optimization technique outperforms in achieving minimum of the specified objective under different network contingencies when compared with evolutionary programming method.

Table 8 presents the number of lines and number of load buses under different severity categories for the three most severe network contingencies and for all the three objective functions. From the Table 8 it is evident that the proposed methods were able to alleviate the overloads with objective functions 2 and 3 only and were unable to alleviate the line overloads with objective function-1 i.e generation cost objective function. This shows the effectiveness of the objective function 2 and 3 for overload alleviation.

Table 8 Number of lines/buses under different severity categories after optimization

Outage Line

Objective Function

Method

Line Loadings

Bus Voltage Profiles

Bus Voltage Stability Indices

LS

BS

AS

MS

BS

AS

MS

VLS

LS

BS

AS

MS

2-5

Before Optimization

28

8

2

2

0

8

16

24

0

0

0

0

Objective Function 1

EP

29

9

1

1

0

20

4

24

0

0

0

0

PSO

27

11

0

2

0

22

2

24

0

0

0

0

Objective Function 2

EP

32

8

0

0

0

24

0

24

0

0

0

0

PSO

32

8

0

0

0

24

0

24

0

0

0

0

Objective Function 3

EP

34

6

0

0

0

24

0

24

0

0

0

0

PSO

34

6

0

0

0

24

0

24

0

0

0

0

11-13

Before Optimization

31

7

0

2

0

8

16

24

0

0

0

0

Objective Function 1

EP

31

7

2

0

0

21

3

24

0

0

0

0

PSO

31

7

1

1

0

19

5

24

0

0

0

0

Objective Function 2

EP

36

4

0

0

0

21

3

24

0

0

0

0

PSO

36

4

0

0

0

22

2

24

0

0

0

0

Objective Function 3

EP

36

4

0

0

0

24

0

24

0

0

0

0

PSO

37

3

0

0

0

24

0

24

0

0

0

0

8-11

Before Optimization

33

4

1

2

0

8

16

24

0

0

0

0

Objective Function 1

EP

33

6

1

0

0

21

3

24

0

0

0

0

PSO

32

7

0

1

0

23

1

24

0

0

0

0

Objective Function 2

EP

35

5

0

0

0

23

1

24

0

0

0

0

PSO

35

5

0

0

0

23

1

24

0

0

0

0

Objective Function 3

EP

36

4

0

0

0

23

1

24

0

0

0

0

PSO

36

4

0

0

0

23

1

24

0

0

0

0

From the above Table 8, it is also observed that the cost objective function may result in reduced objective function value but it was unable to alleviate overloads under contingency condition. The objective function-2 has alleviated the line overloads with the increase in the cost of generation considerably when compared with objective function-1, but it was so effective in alleviating the line overloads by reducing the NCOSI severity index. During minimization of the third objective function, improvement in the voltage profile, voltage stability index and alleviation of the line over loads can be observed by means of drastic decrement in the NCOSI based severity index.

From the Tables 7, it can also be observed that, the PSO and EP methods are able to alleviate the overloads effectively with the NCOSI objective function-3 compared to cost based objective function-1. Based on the observed results for the test system it can be mentioned that the PSO algorithm with NCOSI objective gives more promising results than line flow based and cost based objective function.

Figures 4 shows the percentage line loadings, load bus voltages and voltage stability indices after the optimization by EP and PSO methods with the three objective functions under the rank-1 contingency condition. From the Figures 4, it can be observed that line flows are within their permissible limits during minimization of objectives functions 2 & 3. But line flow violations are observed during minimization of objective function-1(cost of generation) even though cost of generation has been decreased considerably when compared with objective functions 2 & 3.

A comparison between EP and PSO methods with respect to load bus voltage profiles and voltage stability indices is given in the form of bar graphs from the Figures 4 under rank-1 contingency cases for three objective functions. From the Figures 4 it is observed that, efficiency of PSO algorithm is high in maintaining good load bus voltage profiles when compared with EP algorithm.

It can also be noted that during minimization of the NCOSI objective function, the number of lines in most severe category is reduced and there by improvement is observed in the security of the system. Also from the Figures 4, it can be observed that the PSO method with FLCC based objective function is able to maintain good load bus voltage profiles and voltage stability indices under the selected network contingency conditions.

Figure 4(a) Line loadings under rank-1 contingency (objective function-1)

Figure 4(b) Line loadings under rank-1 contingency (objective function-2)

Figure 4(c) Line loadings under rank-1 contingency (objective function-3)

Figure 4(d) Load bus voltages under rank -1 contingency (objective function-1)

Figure 4(e) Load bus voltages under rank-1 contingency (objective function-2)

Figure 4(f) Load bus voltages under rank-1 contingency (objective function-3)

Figure 4(g) Voltage stability indices under rank-1 contingency (objective function-1)

Figure 4(h) Voltage stability indices under rank-1 contingency (objective function-2)

Figure 4(i) Voltage stability indices under rank-1 contingency (objective function-3)

Figure 4 Comparison between EP and PSO methods with respect to load bus voltage profiles and voltage stability indices

8 CONCLUSION

In this chapter, the voltage stability indices at the load buses were also used as the post-contingent quantities, in addition to real power loadings and bus voltages, to evaluate the Fuzzy Logic Composite Criteria based contingency ranking. These post-contingent quantities are expressed in fuzzy set notation. Then the fuzzy rules employed in contingency ranking are compiled to reach the overall system severity index. The proposed contingency ranking method eliminates the masking effect effectively. The proposed fuzzy approach has been tested on large networks and results indicate a clear ranking of contingencies.

The application of EP/PSO methods for solving optimal power flow problems under the selected most severe network contingencies has been presented. The line overloads were relieved through adjustment of generator outputs, generator voltages, tap changing transformers, and shunt compensation. The simulation results on IEEE 30-bus system have been presented for illustration purpose. The algorithms EP and PSO were accurately and reliably converged to the global optimum solution in each case. Moreover, the PSO-algorithm is capable of producing better results compared with other algorithms.

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