Security Constrained Optimal Power Flow Computer Science Essay

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Abstract OPF is a highly non-linear, non-convex, large scale static optimization problem due to large number of variables limit constraints. OPF problem is the perfect incorporation of the contradictory doctrines of maximum economy, safer operation and augmented security.

In any power system, unexpected outages of lines or transformers occur due to faults or other disturbances. These events, referred to as contingencies, may cause significant overloading of transmission lines or transformers, which in turn may lead to a viability crisis of the power system. The principal role of power system control is to maintain a secure system state, i.e., to prevent the power system, moving from secure state into emergency state over the widest range of operating conditions. Security Constrained Optimal Power Flow (SCOPF) is major tool used to improve the security of the system.

In this paper, Genetic algorithm has been used to solve the OPF and SCOPF problems. As initial effort conventional GA (binary coded) based OPF and SCOPF has been attempted. The difficulties of binary coded GA in handling continuous search space lead to the evolution of real coded GAs. Solutions obtained using both the algorithms are compared. Results show that real coded GA is more efficient compared to binary GA in handling OPF and SCOPF problems.

Case studies are made on the IEEE 30 bus test system to demonstrate the ability of real coded GA in solving the OPF and SCOPF problems.

Keywords- Optimal Power Flow (OPF), Power System Security, Genetic Algorithm (GA).

Introduction

Modern Control Centers of electrical power systems are equipped with computational tools to help the operators in their daily work in order to achieve a high quality service with a minimum number of supply interruptions and at a minimum cost. The operation is to be done in a way to maintain the system in a secure mode, i.e., ensuring that the system will be operating continually even when components of the network fail.

The electric system is monitored by the Supervisory Control and Data Acquisition (SCADA) System, which periodically acquires analog measurements and status of switching devices from the network. The monitoring system also allows the operator to act in the system through remote controls, changing switches status and position of transformers tap, etc.

Security Constrained Optimal Power Flow (SCOPF) is a valuable tool in obtaining the optimum way of dispatching a load demand while maintaining system security. The SCOPF has the objective to determine a feasible point of operation that minimizes an objective function, guaranteeing that even if a contingency occurs, the post-contingency state will also be feasible, i.e., without limits violations.

The SCOPF problem has been solved using classical optimization techniques like Gradient method [1] and Newtons method. However, these techniques have difficulty in attaining the global optimum value because of the large number of control variables involved and the discrete nature of the search space. Recent interest has been in solving the problem using evolutionary algorithms like Genetic Algorithm to overcome the problems encountered by the classical techniques.

Whereas Optimal Power Flow (OPF) requires the calculation of the controllable system parameters in such a way that a particular load demand is met in an optimal fashion, SCOPF, on the other hand requires the solution to be optimal in such a way that a single outage of any single system element will not lead to infeasible operating states. This greatly increases the dimension of the problem.

As a preliminary attempt in solving the SCOPF problem, OPF problem has been solved using Genetic Algorithm. A simple GA uses binary coded decision variable to explore the search space. However this approach has difficulties in handling problems with continuous search space. This lead to the evolution of a new version of GA in which the decision variables are real coded in nature. One such algorithm is Real Coded Genetic Algorithm. The effectiveness of the real coded GA is tested against the simple GA in terms of solution quality and computational efficiency using IEEE 30 bus test system.

This solution of the basic OPF problem does not guarantee steady state security. It may be possible that the operating condition which satisfies the desired objective function, violates operating limits under contingencies. It is however, computationally infeasible to work on all the contingencies. Also, some of the contingencies may be of very less or insignificant severity to the operation of the system. Hence, a preliminary study is carried out to evaluate the severity of each contingency in the system. These studies are called contingency studies.

A contingency is said to be more severe if it leads to more number of limit violations or large violations in small number of variables. A severity index is used to calculate the severity of each contingency. Based on this index, the contingencies are ranked in decreasing order of severity. This is called contingency ranking. Depending upon the computational facilities and the tolerances of the system equipment for contingency, a list of credible contingencies is prepared.

The base case OPF problem is then extended to solve for all the credible contingencies and each violation of these contingencies is penalized appropriately using GA. The solution obtained will be optimal in such a way that if any of the listed credible contingencies occur, the system is still in a feasible operating state. The effectiveness of GA in solving the SCOPF problem is evaluated using the IEEE 30 bus test system and the results are compared in solution quality against the results in [1] and [3]. The same problem is once again solved using real coded GA and compared in solution quality and computational efficiency against the result obtained using simple GA..

GENETIC ALGORITHMS

Genetic Algorithm (GA) was modeled & developed by John Holland at University of Michigan in 1960s. It is one kind of direct search algorithms based on the mechanics of natural selection & natural genetics. In nature, individuals in a population compete with each other for various kinds of sources such as food, shelter, & water. Those individuals, who have stronger existing abilities, can survive (survival of fittest) & have relatively larger number of offspring. Conversely, poorly performing ones have less chance to survive & they will produce less or even no offspring at all. This means that highly adapted genes will spread to an increasing number of individuals of the later generation. The combination of good characteristics from different ancestors will probably produce offspring whose fitness is better than that of either ancestor. Finally species will evolve to be more & more suitable to the environment.

Simple Genetic Algorithm

Simple GA uses binary strings to encode the decision variables. Depending upon the required accuracy of the solution, the number of bits in a string is decided. For a typical multi variable optimization problem, all the binary strings of decision variables with different sizes are concatenated to form a bigger string. Each such string is called a 'chromosome. Each chromosome in GA represents a point in the search space. A number of such chromosomes are randomly generated in the first phase of the genetic algorithm. This is called 'Initialization. Together, the chromosome set is called 'Population. The population is evaluated through various operators of GA to generate a new population. This process is carried out until global optimum point is reached. Typically simple GA consists of three phases:

i) Generation, ii) Evaluation and iii) Genetic operation

Real Coded Genetic Algorithm

When binary coded GAs need to be used to handle problems having a continuous search space, a number of difficulties arise. One difficulty is the hamming cliffs associated with certain strings (such as strings 01111 and 10000) from which a transition to a neighboring solution (in real space) requires the alteration of many bits. Hamming cliffs present in a binary coding cause artificial hindrance to a gradual search in the continuous search space. The other difficulty is the inability to achieve any arbitrary precision in the optimal solution.

In binary coded GAs, the string length must be chosen a priori to enable GAs to achieve a certain precision in the solution. The more the required precision, then the larger is the string length. For large strings, the population size requirement is also large, thereby increasing the computational complexity of the algorithm. Since a fixed coding scheme is used to code the decision variables, variable bounds must be such that they bracket the optimum values. Since in many problems this information is not usually known a priori, this may cause some difficulty in using binary coded GAs in such problems.

There exist a number of real parameter GA implementations, where crossover and mutation operators are applied directly to real parameter values. One such method is Mixed Integer Genetic Algorithm (MIGA). Since real parameters are used directly (without any string coding), solving real parameter optimization problems is a step easier when compared to the binary coded GAs. Unlike in the binary coded GAs, decision variables can be directly used to compute the fitness values. Since the selection operator works with the fitness value, any selection operator used with binary coded GAs can also be used in real parameter GAs.

However, the difficulty arises with the search operators. In the binary coded GAs, decision variables are coded in finite length strings and exchanging portions of two parent strings easier to implement and visualize. Simply flipping a bit to perform mutation is also convenient and resembles a natural mutation event. In real parameter GAs, the main challenge is how to use a pair of real parameter decision variable vectors to a mutated vector in a meaningful manner. As in such cases the term 'crossover is not that meaningful, they can be best described as blending operators. However, most blending operators in real parameter GAs are known as crossover operators.

Many crossover operators have been proposed for the real coded genetic algorithms. Some of them are Linear crossover, Naïve crossover, Blend crossover. Similarly many mutation operators like Random mutation, Non-uniform mutation, Normally distributed mutation have are also used. Blend crossover and Random mutation are used in the solution to both OPF and SCOPF using real coded genetic algorithm and they are discussed here.

OPTIMAL POWER FLOWS

The objective of optimal power flow is to find the correct combination of controllable system variables such as generator bus voltage and power output, transformer taps, shunt admittances etc., in such a way that for a given load demand, generating cost is minimal. The OPF might include other constraints such as interface limits and other decisions such as the optimal flow on DC lines and phase shifter angles.

Optimal Power Flow (OPF) has been widely used in power system operation and planning. The Optimal Power Flow module is an intelligent load flow that employs techniques to automatically adjust the Power System control settings while simultaneously solving the load flows and optimizing operating conditions with specific constraints. Optimal Power Flow (OPF) is a static nonlinear programming problem which optimizes a certain objective function while satisfying a set of physical and operational constraints imposed by equipment limitations and security requirements. In general, OPF problem is a large dimension nonlinear, non-convex and highly constrained optimization problem.

Problem Formulation

The standard OPF problem can be written in the following form

Minimize F(x) (Objective function)

Subject to: hi(x)=0, i = 1, 2,......., m (Equality constraints)

g(i) (x)≤0 j= 1, 2,.....,n (Inequality constraints)

There are m- equality constraints and n- inequality constraints and the number of variables is equal to the dimension of the vector x.

Objective function:

The objective function for the OPF reflects the costs associated with generating power in the system. The quadratic cost model for generation of power will be utilized

(1)

Where PGi is the amount of generation in megawatts at generator i. The objective function for the entire Power System can then be written as the sum of the quadratic cost model at each generator.

The total fuel cost for an ng-generator system is calculated as

(2)

This objective function will minimize the total system costs, and does not necessarily minimize the costs for a particular area within the Power System.

If the valve-point loading effect of thermal units is also taken into consideration, the fuel cost of a generator will be of the form

(3)

Where i= 1,2,….ng.

ai, bi, ci, di are the cost coefficients of the unit i.

The sinusoidal term added to the fuel cost function which models the valve-point effect introduces ripples to heat-rate curve and therefore introducing more local minima to the search space. Genetic algorithm, being a heuristic search technique, will not have any difficulties in handling such systems. Quadratic cost functions have been assumed for the generators in the thesis.

Equality constraints:

The equality constraints of the OPF reflect the physics of the Power System as well as the desired voltage set points throughout the system. The physics of the Power System are enforced through the power flow equations which require that the net injection of real and reactive power at each bus sum to zero

(4)

(5) Inequality constraints:

The inequality constraints of the OPF reflect the limits on physical devices in the Power System as well as the limits created to ensure system security. Physical devices that require enforcement of limits include generators, tap changing transformers, and phase shifting transformers. This section will lay out all the necessary inequality constraints needed for the OPF implemented in this thesis.

Generators have maximum and minimum output powers and reactive powers which add inequality constraints.

(6) (7)

Load tap changing transformers have a maximum and a minimum tap ratio which can be achieved and shunt admittance limits of switchable capacitor/reactor devices have a maximum and a minimum limit, which can be achieved. Both of these create inequality constraints.

(8)

(9)

For the maintenance of system security, Power Systems have transmission line as well as transformer MVA ratings. These ratings may come from thermal ratings (current ratings) of conductors, or they may be set to a level due to system stability concerns. The determination of these MVA ratings will not be of concern in this thesis. It is assumed that they are given. Regardless, these MVA ratings will result in another inequality constraint.

, k Є (10)

To maintain the quality of electrical service and system security, bus voltages usually have maximum and minimum magnitudes. These limits again require the addition of inequality constraints.

(11) Variables:

Practically the variables in OPF problem can be divided into two as continuous variables and discrete variables.

Continuous variables are active power generation of generators (𝑃𝑔) and Generator bus voltage magnitudes (V).

Discrete Variables are transformer tap setting (𝑇𝑝), VAR injection values of switchable shunt capacitor /reactor (𝑌ℎ) and phase shifter angle positions.

In practical operation, because the reactive power injection of switched shunt device depends on the bus voltage, the specified VAR-injection value of switched shunt device (such as shunt capacitor bank) is difficult to adjust. A solution to the problem is to select the switched shunt admittance as the discrete control associated with the switched shunt devices; it will be capable of obtaining good control level.

By adding the security constraints, the objective function, OPF problem is expressed as a mixed integer non-convex programming.

Algorithm of OPF using GA

1. Read input data

2. Form Y-Bus using sparsity technique

3. Initialize random population and set generation count gen=1

4. If gen>genmax goto step 14, else goto step 5

5. Initialize chromosome count ii=1.

6. If chromosome count ii<psize, goto step 7, else increment generation count (gen=gen+1) and goto step 4

7. Decode the chromosome and determine the actual control variables

8. Modify the Y-Bus depending on the control variables and run NR load flow

9. Compute the fuel cost and check all the constraints such as bus voltage limits, line power transfer limit, generator reactive power limit, slack generator active power limit. If the NR load flow did not converge, assign a very high value as fuel cost

10. Determine the violated constraints and compute the associated penalty cost

11. Calculate the fitness of the chromosome

Fit(ii) = K/(fuel cost+ penalty cost)

12. Arrange the chromosomes and their fitness values in descending order of fitness. Check for convergence. If converged goto step 14, else goto step 13

13. Apply GA operators and generate new population. Increment chromosome count (ii=ii+1), goto step 6

14. Maximum number of generations over. Print results

CONTINGENCY ANALYSIS AND RANKING

Contingency analysis and ranking is the process in which various contingencies like line outage, generator outage, transformer outage etc., are simulated and ranked in descending order of severity. This is an offline study and is useful to check and improve the security of the system. Outage studies are usually approximate studies. The accuracy levels for convergence of load flows can be 0.001, 0.0005, 0.01 p.u. For rigorous calculations, 0.0001 is used.

Rigorous outage studies will be carried out for planning and expansion studies as offline. For online studies, most of the time approximate studies are used. For smaller systems, rigorous studies can be preferred without going for contingency analysis and ranking. For large size power systems, contingency ranking and analysis is used to prepare a list of contingencies which are more severe. A line outage is said to be more severe if it causes more number of bus voltage drops and a large number of lines are overloaded because of the outage.

Various sensitivity factors are used to study line and generator outages. To save the computation time and memory requirement resistance of all elements are neglected so that sensitivity factors will be real coefficient matrices. These sensitivity factor matrices are usually calculated offline and stored as lookup tables for online studies. In this work, contingency analysis and ranking is done offline using rigorous NR Q-adjusted load flow studies with an accuracy of 0.0001 p.u.

During the formation of Y-Bus matrix, transformers are represented in the equivalent π representation. Also, the transmission lines are represented in equivalent π representation. Hence, both transformers and transmission lines are treated in the same way.

Contingency selection involves the selection of lines or generators whose outage is more severe. To identify the severity of a transmission line, there is no specific approach by which a unique solution can be obtained. Different methods are suggested for identifying the severity. One such index used in [2] is discussed here

Severity Index:

For a line outage 'k, the severity index is defined as:

(12) where, SI = Severity Index (Overload index)

Sl =MVA flow in line l

Slmax=MVA rating of line l

L =set of overloaded lines

m =integer exponent

Based on the severity index assigned to each line outage, a list is prepared. This is done by first arranging the lines in the descending order of their severity and taking the first few lines with the highest severity. For a large power system, 5 to 10% of the lines can be chosen in the contingency list. It is assumed that since these are the more severe outages in the system, handling them in the SCOPF will be fairly enough to improve the security of the system. A value of m=1 has been used.

The line outage simulation study can be summarized in the following steps:

1. Choose the line number 'k for which line outage is to be simulated.

2. The sending and receiving ends of the line 'k are stored in 'p and 'q respectively.

3. Take R(k)=10^20 p.u., X(k)=10^20 p.u. and Ycp(k)=0, Ycq(k)=0 Line outage is simulated.

4. Four locations of Y- Bus i.e., Ypp, Yqq, Ypq, Yqp are modified.

5. Load flow is run for the system with modified Y- Bus assuming the converged base case voltages as initial guess voltages. Voltages and phase angles for all buses are obtained.

6. Calculate the line flows and severity index.

7. Restore the Y- Bus and the resistance and reactance of the line to their original values.

SECURITY CONSTRAINED OPTIMAL POWER FLOWS

In any power system, unexpected outages of lines or transformers occur due to faults or other disturbances. These events, referred to as contingencies, may cause significant overloading of transmission lines or transformers, which in turn may lead to a viability crisis of the power system. The principal role of power system control is to maintain a secure system state, i.e., to prevent the power system, moving from secure state into emergency state over the widest range of operating conditions. Security Constrained Optimal Power Flow (SCOPF) is major tool used to improve the security of the system.

The security of the system can be improved either through preventive control or post contingency corrective action. Alsac and Stott [1] extended the penalty function method to security constrained optimal power flow problem in which all the contingency case constraints are augmented to the optimal power flow problem. In this method the functional inequality constraints are handled as soft constraints using penalty function technique. The drawback of this approach is the difficulty involved in choosing proper penalty weights for different systems and different operating conditions which if not properly selected may lead to excessive oscillatory convergence. This combined with prohibitively large computing time makes this method unsuitable for online implementation.

Apart from using preventive approach for security enhancement, the post contingency state corrective action can also be used for security enhancement. The resulting stage has the same security level as the usual security - constrained optimal power flow case with lower operating cost. The power electronics-based FACTS devices can also be employed for corrective action due to its high speed of response.

The proposed algorithm solves the SCOPF problem subject to the power balance equality constraints, limits on control variables namely active power generation, controllable voltage magnitude pertaining to the base case and selected contingency cases. The effectiveness of the proposed approach is demonstrated through preventive and corrective control action for a few harmful contingencies in the IEEE -30 bus system.

Problem Formulation

The objective of the SCOPF problem is the minimization of total fuel cost pertaining to base case and alleviation of line over load under contingency case. The adjustable system quantities such as controllable real power generations, controllable voltage magnitudes, controllable transformer taps are taken as control variables. The equality constraint set comprises of power flow equations corresponding to the base case as well as the postulated contingency cases. The inequality constraints include control constraints, reactive power generation and load bus voltage magnitude and transmission line flow constraints pertaining to the base case as well as the postulated contingency cases. The mathematical description of objective functions and its associated constraints are presented below. For each individual, the equality constraints (3.3) and (3.4) are satisfied both in base case as well as contingency cases by running NR algorithm and the constraints on the state variables are taken into consideration by adding penalty function to the objective function.

(13)

where, represents the total fuel cost,

represents the severity index for outage l,

,, and are the penalty terms for the reference bus generator active power limit violation, load bus voltage limit violation; reactive power generation limit violation and the line flow limit violation respectively.

These quantities are defined by the following equations:

(14)

(15)

(16)

(17)

GA is usually designed to maximize the fitness function which is a measure of the quality of each candidate solution. Therefore a transformation is needed to convert the objective of the OPF problem to an appropriate fitness function to be maximized by GA. Therefore the GA fitness function is formed as F=k/f, where, 'k is a large constant.

Algorithm of SCOPF using GA

1. Read input data

2. Form Y-Bus using sparsity technique

3. Initialize random population and set generation count gen=1

4. If gen>genmax goto step 16, else goto step 5

5. Initialize chromosome count ii=1.

6. If chromosome count ii<psize, goto step 7, else increment generation count (gen=gen+1) and goto step 4

7. Decode the chromosome and determine the actual control variables

8. Modify the Y-Bus depending on the control variables and run base case NR load flow

9. Compute the fuel cost and check all the constraints such as bus voltage limits, line power transfer limit, generator reactive power limit, slack generator active power limit. If the NR load flow did not converge, assign a very high value as fuel cost

10. Determine the violated constraints and compute the associated penalty cost

11. Simulate a contingency from the list of contingencies, modify Y-Bus appropriately and run NR load flow. Calculate the contingency penalty cost for the violations of constraints in the contingency case.

12. Repeat step 11 for all the contingencies in the list and calculate the total contingency penalty cost

13. Calculate the fitness of the chromosome

Fit(ii) = K/(fuel cost+ penalty cost+ contingency penalty cost)

14. Arrange the chromosomes and their fitness values in descending order of fitness. Check for convergence. If converged goto step 14, else goto step 13

15. Apply GA operators and generate new population. Increment chromosome count (ii=ii+1), goto step 6

16. Maximum number of generations over. Print results

RESULTS AND DISCUSSION

The SCOPF in its general form is a nonlinear, non convex, static, large scale optimization problem with both continuous and discrete variables in large number. As an initial attempt in solving SCOPF using binary and real coded genetic algorithms, OPF problem has first been solved. As the SGA has the difficulty in handling continuous search space, new versions of GAs are evolved. One such algorithm used here is real coded Genetic Algorithm. The effectiveness of real coded GA is compared in solution quality and computational efficiency against simple GA.

The Optimal Power Flow (OPF) is a highly non-linear, large scale optimization problem due to large number of variables & constraints. It has both continuous and discrete variables as its decision variables. OPF with Fuel cost minimization as objective function is formulated as a single objective optimization case.

The algorithm is implemented using Matlab® 2008a and is tested for its robustness on a standard IEEE 30 bus system. The network data is shown in Appendix A. The network consists of 6 Generator buses, 21 load buses & 41 lines, of which 4 lines are due to tap setting transformers. The total load on the network is 283.4 MW. The algorithms have been implemented on a personal computer with 2.44 GHz Intel Pentium 4 processor and 1.2 GB RAM.

Present thesis considers 24 control variables as explained below. Five generator active power outputs, six generator-bus voltage magnitudes, four transformer tap-settings & nine shunt susceptances.

OPF with simple Genetic Algorithm

The gene length for unit active power outputs is 12 bits, generator voltage magnitude is 8 bits, and both of them are treated as continuous control variables. As the transformer tap settings can take 17 discrete values each one is encoded using 5 bits & the step size is 0.0125 p.u. The bus shunt susceptance can take 6 discrete values each one is encoded using 3bits, & the step size is 0.01 p.u. (on system MVA basis). Thus, the total string length would be 155.

PARAMETERS: Population size =60, Uniform Crossover Probability =0.85, String length 155 bits, Mutation Probability=0.01 & elitism 0.15 & Maximum Number of iterations=100. Roulette wheel selection technique is used for parent selection.

OPF with Real coded Genetic Algorithm

The control variables considered are similar to the variables explained above

PARAMETERS: Population size =60, Maximum Number of iterations=100, Crossover Probability =0.85, Mutation Probability=0.01, string length = 24 variables & elitism 0.15.

The results obtained for various control variables using simple and real coded GA are given. If the problem did not converge in maximum number of generations, solution obtained at the last generation can be taken as the optimum value.

TABLE I

Optimal settings of control variables for OPF using GA

Variables

Simple GA

Real Coded GA

Slack

181.70

177.60

PG2

43.35

48.04

PG5

21.40

21.36

PG8

20.79

21.45

PG11

13.37

11.79

PG13

12.11

12.21

VG1

1.0758

1.0818

VG2

1.0529

1.0641

VG5

1.0235

1.0345

VG8

1.0319

1.0384

VG11

1.0100

1.0404

VG13

1.0717

1.0415

Tap 6-9

0.9875

1.0250

Tap 6-10

0.9000

0.9375

Tap 4-12

0.9875

0.9750

Tap 28-27

0.9750

0.9875

Shunt 10

0.02

0.05

Shunt 12

0.05

0.03

Shunt 15

0.04

0.02

Shunt 17

0.05

0.05

Shunt 20

0.05

0.01

Shunt 21

0.01

0.02

Shunt 23

0.00

0.03

Shunt 24

0.02

0.04

Shunt 29

0.04

0.05

Optimal fuel cost($/hr)

802.23

800.66

Time

545.45 sec

547.84 sec

It is observed that the results obtained by real coded GA are more optimal than the one obtained using binary GA. Also, the result obtained using mathematical technique reported in [1] i.e., 802.4 $/hr, is nearer to the one obtained using binary GA. This shows that GA works better than the mathematical techniques in finding the optimal solution for the OPF problem. Convergence characteristics for real and binary coded GA are plotted. Voltage profile of all the buses after convergence is shown in bar graph. Red colour in the graph indicates generator bus

Fig 1 Convergence characteristic of simple GA for OPF

Fig 2 Convergence characteristic of real coded GA for OPF

Fig 3 Voltage profile for OPF using binary coded GA

Fig 4 Voltage profile for OPF using binary coded GA

Contingency Analysis And Ranking

Since contingency analysis and ranking is an offline study, rigorous analysis has been done using NR Q-adjusted load flows with a convergence criterion of 0.0001. The results of the analysis are summarized in the table hereunder.

TABLE III

Summary of contingency analysis for IEEE 30-bus system

Outage line No.

Overloaded lines

Line flow

(MVA)

Line flow limit (MVA)

Severity Index

(SI)

Rank

1

1-3

3-4

147.03

140.60

130

130

2.712

1

2

1-2

142.28

130

1.312

3

4

1-2

139.55

130

1.237

4

5

2-6

5-7

67.18

72.77

65

70

2.227

2

25

15-18

16.36

16

1.068

5

From the above table it can be seen that the 25th line outage does not affect the operation of the system adversely as the line is overloaded only to a very small extent. Out of 41 lines, for a 30 bus system, 4 are transformers modeled as transmission lines. Transformer and generator outages are not considered in the present thesis because it is leading to an infeasible operating state and it is observed that nothing can be done to make the system secure under these outage conditions. Full SCOPF and contingency analysis, however, should take into account all the possible outages. This is not possible in this case as the system size is too small to consider the generator and transformer outages.

Security Constrained Optimal Power Flows

The data for the SCOPF program is the same as that for the OPF program, including the GA parameters for both real and binary coded SCOPF. The results obtained using binary and real coded GA are compared in the table hereunder.

TABLE IIIII

Optimal settings of control variables for SCOPF using GA

Variables

Simple GA

Real Coded GA

Slack

148.07

149.25

PG2

59.34

57.49

PG5

26.02

24.69

PG8

31.90

32.98

PG11

13.18

17.57

PG13

18.98

15.51

VG1

1.0545

1.0563

VG2

1.0643

1.0477

VG5

1.0187

1.0211

VG8

1.0351

1.0274

VG11

1.0847

1.0879

VG13

1.0615

1.0431

Tap 6-9

1.0250

1.0625

Tap 6-10

1.0000

1.0250

Tap 4-12

0.9875

1.0625

Tap 28-27

0.9625

0.9750

Shunt 10

0.02

0.05

Shunt 12

0.05

0.03

Shunt 15

0.04

0.02

Shunt 17

0.05

0.05

Shunt 20

0.05

0.01

Shunt 21

0.01

0.02

Shunt 23

0.00

0.03

Shunt 24

0.02

0.04

Shunt 29

0.04

0.05

Optimal fuel cost($/hr)

813.56

811.41

Time

1546 sec

987 sec

While solving SCOPF using simple GA, it is observed that the solution obtained is not consistently obtained over a large number of trails. The value of fuel cost obtained as the part of final solution varied from 812 to 818 $/hr. This when compared to the value obtained using mathematical technique in [1], i.e., 813.74 $/hr does not fare better. The value of fuel cost obtained using real coded Genetic Algorithm varied from 810 to 813 $/hr over a large number of trails and is better compared to the value reported in [1].

Here, it is to be observed that though some author reported a better value for SCOPF using genetic algorithm [2], it is because the penalty for voltage violations in the contingency cases has not been considered. Simple GA does not appear to be competent in handling a large number of penalty functions added to the basic objective function. Real coded GA, on the other hand worked better for both OPF and SCOPF. Given hereunder are the graphs of best fuel cost obtained for each generation in binary and real coded genetic algorithms.

It is observed that there are no violations in base case or contingency cases in the solution obtained by using real or binary coded genetic algorithms. The generator bus voltages in the bar graph that follows are given in red and load bus voltages in blue. Similarly the bus limits are marked by straight lines in the bar graph.

Fig 5 Convergence characteristic of binary coded GA for SCOPF

Fig 6 Convergence characteristic of real coded GA for SCOPF

Fig 7 Voltage profile for SCOPF using binary coded GA

Fig 8 Voltage profile for SCOPF using real coded GA

CONCLUSION

In this paper, OPF problem is first attempted using binary and real coded Genetic Algorithms. Contingency analysis and ranking is done to find the most severe line outages. These severe contingencies are used in the solution of SCOPF problem using penalty factor method. All the violations in contingency cases are added to the base case fuel cost as penalty. The SCOPF problem is also attempted using binary and real coded Genetic Algorithms.

Case studies for all the algorithms are made on the standard IEEE 30 bus test system. Based on the investigations carried out at various stages of the thesis and presented in different chapters, the following conclusions can be drawn.

Binary coded GA works on par with the classical mathematical techniques for solving the OPF problem. The average solution obtained using binary GA over a large number of trial runs is about 802.3 $/hr, which is nearly same as the one reported in [1], i.e., 802.4$/hr, obtained using mathematical technique (Dommel- Tinney approach).

Real coded GA works better in finding the global optimal solution for the OPF problem when compared with the binary coded GA. The average solution obtained using real coded GA is about 801.4$/hr, which is a better value compared to the above mentioned values obtained using binary coded GA or the mathematical technique.

The average time taken for convergence of real coded GA is less compared to that of the binary coded GA for the OPF problem. The average time taken for ten trial runs of the real coded GA is nearly 300 seconds. This is better compared to the average of about 500 seconds for the binary coded GA.

Binary coded GA does not appear to be competent enough to handle the SCOPF problem. The solution obtained for SCOPF using binary GA showed a wide variation over a large number of trial runs. The value of fuel cost varied from 812 $/hr to 818 $/hr over ten trial runs. The average solution obtained does not fare better compared to the value 813.74$/hr, obtained using mathematical technique reported in [1].

Real coded GA works better than the mathematical approach for solving SCOPF problem. This is proved by the quality of the solution. The result varied from 810 $/hr to 813 $/hr over ten trial runs. This is a better average value compared to the above mentioned values obtained using binary coded GA or the mathematical technique.

The average time taken for convergence is about 1100 seconds for solving the SCOPF using real coded GA. The time taken for convergence increases as the number of contingencies included in the contingency list of SCOPF increases.

The quality of solution achieved & the speed with which it is attained are greatly influenced by the load flow technique used for solution of equality constraints & the optimization technique used for modifying the control variables.

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