With the emerging trend of restructuring in the electric power industry, many transmission lines have been forced to operate at almost their full capacities worldwide. Due to this, more incidents of voltage instability and collapse are being observed throughout the world leading to major system breakdowns. To avoid these undesirable incidents, a fast and accurate estimation of voltage stability margin is required. In this paper, genetic algorithm based back propagation neural network (GABPNN) has been proposed for voltage stability margin estimation which is an indication of the power system's proximity to voltage collapse. The proposed approach utilizes a hybrid algorithm that integrates genetic algorithm and the back propagation neural network. The proposed algorithm aims to combine the capacity of GAs in avoiding local minima and at the same time fast execution of the BP algorithm. Input features for GABPNN are selected on the basis of angular distance-based clustering technique. The performance of the proposed GABPNN approach has been compared with the most commonly used gradient based BP neural network by estimating the voltage stability margin at different loading conditions in 6-bus and IEEE 30-bus system. GA based neural network learns faster, at the same time it provides more accurate voltage stability margin estimation as compared to that based on BP algorithm. It is found to be suitable for online applications in energy management systems.

Voltage stability is concerned with the ability of the power system to maintain acceptable voltages at all the system buses under normal conditions as well as after being subjected to a disturbance. Thus, the analysis of voltage stability deals with finding the voltage levels at all buses in the system under different loading conditions to ascertain the stability limit and margin. A power system enters a state of voltage instability when a disturbance, increase in load demand or change in system conditions causes a progressive and uncontrollable decline of bus voltages.

The main factor causing voltage instability is the inability of the power system to meet the reactive power demand. In most of the cases, voltage profiles show no abnormality prior to undergoing voltage collapse because of the load variation. Voltage stability margin (VSM) is a static voltage stability index which is used to quantify how “close” a particular operating point is to the point of voltage collapse [

During the last few years, several methodologies for detecting the voltage collapse points (saddle-node bifurcations) in power systems using steady-state analysis techniques have been modified and applied for the determination of analyzing voltage stability of power systems for example PV and QV curves, sensitivity-based indices [

These analytical methods involve considerable computational effort and require significantly large computational time and, hence, cannot be used directly for online monitoring and initiation of preventive control actions to enhance system voltage stability. For online applications, there is a need for quick detection of the potentially dangerous situations of voltage instability so that necessary actions may be taken to avoid the occurrence of voltage collapse in a power system.

Recently, artificial neural networks (ANNs) have been proposed for voltage stability evaluation [

In typical power systems, there are voluminous amount of input data. Then, the success of ANN applications also depends on the systematic approach of selecting highly important features which will result in a compact and efficient ANN. Different feature reduction methods for voltage stability assessment are compared in [

Voltage instability is, in general, caused by either of two types of system disturbances: increase in load demand and contingencies. In the present paper, voltage instability due to increase in load demand is considered. A genetic algorithm-based back propagation neural network [

These conventional methods of voltage stability assessment are computationally intensive and data sensitive. On the other hand, artificial neural network-based approach is fast and provides result even with partially missing/noisy data. Back propagation (BP) searches on the error surface by means of the gradient descent technique in order to minimize the error. It is therefore likely to get struck in a local minimum [

The idea to hybridize the two approaches, namely, GA and BPN follows naturally. Rajasekaran and Pai [

Genetic algorithm is an adaptive search technique used for solving mathematical problems and engineering optimization problems that emulates Darwin’s evolutionary theory that is fittest is likely to survive. Genetic algorithm attempts to find a good (or best) solution to the problem by genetically breeding a population of individuals over a series of generations. In genetic algorithm, each individual in the population represents a candidate solution to the given problem. The GA transforms a population (set) of individuals, each with an associated fitness value, into a new generation of the population using reproduction, crossover and mutation. Core of the GA is genetic recombination of strings. Generally, a population of strings is randomly generated at the beginning of the process.

An important characteristic of GA is that global feature of search is related to the diversity of the initial population: the more diverse the population, the more global the search. From the initial population, selection strategy based on fitness proportion is adopted to select individuals in current population. Higher selective pressure often leads to the loss of diversity in the population, which causes premature convergence but at the same time improves convergence speed. Therefore, a balance is required between population diversity and convergence speed for obtaining the good performance of GA. Then reproduction, cross-over, and mutation operators are randomly applied to produce next generation population until genetic stopping condition is satisfied.

When the GA is correctly implemented for solving any problem, the population evolves over successive iterations with the fitness value increasing towards global optimum. Several features of GAs like no dependency on gradient information, less likely to be trapped in local minima, ability to deal with the problems where no explicit/exact objective function is available, ability to deal with the concave objective function-based optimization problems, and make them much more robust than many other search algorithms. Moreover GAs are much superior to conventional search and optimization techniques in high-dimensional problem space due to their inherent parallelism and directed stochastic search implemented by recombination operators. GA operates[

Genetic algorithm operations.

(A) Generate randomly an initial population of individuals. | |

(B) Carry out the following substeps iteratively for each generation until a termination condition is fulfilled. | |

(i) Evaluate fitness of each individual to check its ability to solve the specific problem and save the best individual of all preceding population. | |

(ii) Select pair of individuals to be parents for reproduction on the basis of their fitness. | |

(iii) Generate offsprings from parents by implementing genetic search operators such as cross-over/mutation. Add them to the population. |

As shown in Table

Increasing the crossover probability increases the recombination of building blocks. But it also increases the disruption of good strings.

Increasing the mutation probability tends to transform the genetic search into random search, but it also helps reintroduce lost genetic material.

Increasing the population size increases the diversity and reduces the probability of premature convergence to a local optimum, but it also increases the time required for the population to converge to the optimal region in the search space.

Artificial neural networks and genetic algorithms are both abstractions of natural processes. They are formulated into a computational model so that the learning power of neural networks and adaptive capabilities of evolutionary processes can be combined [

The GA-based weight optimization during training of an ANN follows two steps. The first step is encoding strings for the representation of connection weights. The second step is the evolutionary process simulated by GA, in which search operators have to be implemented in conjunction with the representation scheme. The evolution stops when the population has converged. A population is said to have converged when 95% of the individuals constituting the population share the same fitness value [

General framework of GAs for neural network training.

(i) Decode each individual in the current population into a set of connection weights and construct a corresponding ANN with the weights. | |

(ii) Evaluate the ANN by computing its total mean square error between actual and target outputs. | |

(iii) Determine fitness of individual as inverse of error. The higher is the error, the lower is the fitness. | |

(iv) Store the weights for mating pool formation. | |

(v) Implement search operators such as cross-over/mutation to parents to generate offsprings. | |

(vi) Calculate fitness for new population. | |

(vii) Repeat steps (iii) to (vi) until the solution converge. | |

(viii) Extract optimized weights. |

Obstacles to the success of GA in evolving the weights for a fixed network structure include the manner in which weights are encoded to the chromosomes and the definition of the “fitness function” that allows the preferential reproduction of good offsprings and prevents premature convergence to a poor solution [

To determine the fitness value for each of the chromosomes, we extract weight from each of chromosomes.

Let

In any neural network application, if a large number of input variables are used, the number of interconnection weights will increase and the training of neural network will be extremely slow. To overcome this problem, those variables are selected as input to a neural network that has significant effect on its output, that is, on voltage stability margin. Performance of any neural network mainly depends upon the input features selected for its training. It is essential to reduce the number of inputs to a neural network and to select its optimum number.

To select the input features, angular distance base-clustering method is used. The basic principle for clustering is to group the total

With the state vector

It is observed that row

Those system variables with similar vector

Vectors

This cosine value can be used to evaluate the degree of similarity between two vectors. If

Let the system variable

Increase

Compute the cosine value

If

If

If vector

If vector

Create a new cluster

Increase

If

If there is any move in cluster elements in the preceding

For each cluster

During training of a neural network, the higher valued input variables may tend to suppress the influence of smaller ones. To overcome this problem, the neural networks are trained with normalized input data, leaving the network to learn weights associated with the connections emanating from these inputs. The raw data are scaled in the range 0.1–0.9 for use by neural networks to minimize the effect of magnitude between input [_{n}

Sequential steps (flowchart) for developing GABPNN proposed for voltage stability margin estimation are illustrated in Figure

Flow chart for VSM estimation using GABPNN.

Generate a large number of load patterns by perturbing the loads at all the buses in wide range randomly.

Normalize input data as selected from the angular distance base-clustering method and the output, that is, voltage stability margin

Set numbers of generations for genetic optimization of weights.

Initialize structure for the neural network, that is, input-hidden-output nodes for determining the number of weights.

Set generation count as

Generate randomly the initial population

Extract weights for each of the population

Calculate the fitness value for each individual in the population as,

Get the mating pool ready by replacing worst fit individuals with high-fit individuals.

Using cross-over mechanism, reproduce offsprings from the parent chromosomes.

Next generation population is achieved. Increase generation count by 1, that is,

Check, if

Training is complete. Extract optimized weights from converged population

Test the developed GABPNN for unseen load patterns.

In the proposed GABPNN, GA performs the weight adaptation for acquiring the minimized error during the training. Before executing certain task, GA requires several parameters to be devised for its proper functioning. Some of them are gene encoding, population initialization, selection, reproduction, fitness evaluation, and so forth. The basic computing elements in GAs are

There is no definite view in the literature about suitability of encoding schemes for chromosomes. Too complicated scheme provides high flexibility in problem representation but may reduce GA’s efficiency due to complicated operations. On the other hand, too simple representation may suffer from slow or premature convergence. This requires a careful selection of an encoding scheme such that GA operations are not compromised but still provides enough flexibility to support dynamic weight adaptation. In the present work, real value coding is adopted for

Size of the population of individuals (chromosomes) is generated randomly to start the genetic search procedure. The size of population depends upon number of weights to be optimized multiplied by

Thus, population size depends upon the number of digits to be used to represent each weight. An appropriate gene length is required for an adequate population size. This way selection of digits to represent a weight is of great importance. Too few digits may result in poor convergence, while a large number of digits per weight will lead to slow convergence due to very long chromosomal string. In the present work, each weight is encoded as a fixed 5-digit string (i.e.,

Convergence is the progress towards increasing uniformity in fitness values, as each time lowest fitness is replaced with maximum fitness value. Fitness function is taken as inverse of root mean square error (RMS) function. For the converged population, that is, group of individuals comprising minimum RMS, final optimized weights are extracted and decoded. These optimized weights belong to the trained GABPNN, which is ready for testing on unseen load patterns.

The GA-based back propagation neural network approach has been implemented for voltage stability margin estimation for standard 6-bus system [

Out of 250 patterns, 200 patterns are selected randomly for training and the remaining 50 for testing the performance of the trained GABPNN model. During training, it has been found that the number of hidden nodes has affected the convergence rate by increasing or decreasing the complexity of the neural network architecture. Hence, hidden nodes are selected on the “trial and error” basis for obtaining fitness convergence in the minimum number of generations with high convergence rate. In VSM estimation problem, the optimum size of the GABPNN has been found to be 10-4-1. The optimal training for GABPNN was achieved in 15 iterations only.

During testing phase, the 50 testing patterns were tested for evaluating the performance of the trained GABPNN. The testing results of all the 30 patterns are shown in Table

Output and error during testing of GABPNN (10-4-1).

TP | Voltage stability margin | Error (p.u.) | Error (% age) | TP | Voltage stability margin | Error (p.u.) | Error (% age) | ||

Actual | By GABPNN | Actual | By GABPNN | ||||||

1 | 1.45935 | 1.4304 | 0.029 | 1.9849 | 2 | 1.62284 | 1.6169 | 0.0059 | 0.3612 |

3 | 1.57712 | 1.5733 | 0.0038 | 0.2397 | 4 | 1.41375 | 1.3865 | 0.0272 | 1.9244 |

5 | 1.64237 | 1.6341 | 0.0083 | 0.5061 | 6 | 1.52868 | 1.5105 | 0.0182 | 1.1905 |

7 | 1.55574 | 1.5763 | −0.0206 | −1.3238 | 8 | 1.43946 | 1.4307 | 0.0088 | 0.6145 |

9 | 1.5849 | 1.584 | 0.0009 | 0.0561 | 10 | 1.47208 | 1.5002 | −0.0281 | −1.9118 |

11 | 1.41474 | 1.4116 | 0.0031 | 0.2221 | 12 | 1.52114 | 1.5229 | −0.0018 | −0.1206 |

13 | 1.48234 | 1.5262 | −0.0439 | −2.9627 | 14 | 1.49136 | 1.4847 | 0.0067 | 0.4479 |

15 | 1.51966 | 1.5246 | −0.0049 | −0.3213 | 16 | 1.52114 | 1.5486 | −0.0275 | −1.8086 |

17 | 1.71256 | 1.7131 | −0.0006 | −0.0356 | 18 | 1.58577 | 1.6148 | −0.029 | −1.8276 |

19 | 1.65003 | 1.6502 | −0.0002 | −0.011 | 20 | 1.59516 | 1.595 | 0.0001 | 0.0048 |

21 | 1.45626 | 1.4538 | 0.0025 | 0.1718 | 22 | 1.52262 | 1.5097 | 0.0129 | 0.8503 |

23 | 1.5655 | 1.5485 | 0.017 | 1.0834 | 24 | 1.47739 | 1.4733 | 0.0041 | 0.276 |

25 | 1.55784 | 1.5675 | −0.0096 | −0.6174 | 26 | 1.82254 | 1.8215 | 0.0011 | 0.0619 |

27 | 1.43748 | 1.4152 | 0.0223 | 1.5501 | 28 | 1.44526 | 1.4396 | 0.0057 | 0.394 |

29 | 1.61876 | 1.5927 | 0.0261 | 1.6113 | 30 | 1.68389 | 1.6946 | −0.0107 | −0.6365 |

TP: Testing Pattern.

Testing performance of the trained GABPNN and BPNN.

As the BP neural network is the most popular ANN model and has been implemented in almost every area of engineering, in this paper the performance of GABPNN model has been compared with BPNN model. To compare the effectiveness of the proposed GABPNN approach, a BP model with the same structure that is 10-4-1 was trained for 2500 iterations. The testing results of the trained BPNN are given in Table

Output and error during testing of BPNN (10-4-1).

TP | Voltage stability margin | Error (p.u.) | Error (% age) | TP | Voltage stability margin | Error (p.u.) | Error (% age) | ||

Actual | By BPNN | Actual | By BPNN | ||||||

1 | 1.45935 | 1.6012 | −0.1419 | 9.72028 | 2 | 1.62284 | 1.60577 | 0.01707 | 1.05201 |

3 | 1.57712 | 1.6534 | −0.0763 | 4.83691 | 4 | 1.41375 | 1.59554 | −0.1818 | 12.8586 |

5 | 1.64237 | 1.6064 | 0.03596 | 2.18981 | 6 | 1.52868 | 1.60288 | −0.0742 | 4.85421 |

7 | 1.55574 | 1.59965 | −0.0439 | 2.82266 | 8 | 1.43946 | 1.59497 | −0.1555 | 10.8038 |

9 | 1.5849 | 1.55658 | 0.02833 | 1.78727 | 10 | 1.47208 | 1.59467 | −0.1226 | 8.32796 |

11 | 1.41474 | 1.595 | −0.1803 | 12.7416 | 12 | 1.52114 | 1.56406 | −0.0429 | 2.82164 |

13 | 1.48234 | 1.60372 | −0.1214 | 8.18847 | 14 | 1.49136 | 1.54934 | −0.058 | 3.88758 |

15 | 1.51966 | 1.55096 | −0.0313 | 2.06016 | 16 | 1.52114 | 1.57032 | −0.0492 | 3.23347 |

17 | 1.71256 | 1.60263 | 0.10993 | 6.41887 | 18 | 1.58577 | 1.59772 | −0.012 | 0.75347 |

19 | 1.65003 | 1.58203 | 0.068 | 4.1212 | 20 | 1.59516 | 1.60209 | −0.0069 | 0.43449 |

21 | 1.45626 | 1.59506 | −0.1388 | 9.53137 | 22 | 1.52262 | 1.60296 | −0.0803 | 5.27643 |

23 | 1.5655 | 1.54795 | 0.01755 | 1.12097 | 24 | 1.47739 | 1.60231 | −0.1249 | 8.45509 |

25 | 1.55784 | 1.56032 | −0.0025 | 0.15901 | 26 | 1.82254 | 1.54385 | 0.27869 | 15.2913 |

27 | 1.43748 | 1.60245 | −0.165 | 11.4763 | 28 | 1.44526 | 1.58866 | −0.1434 | 9.92211 |

29 | 1.61876 | 1.59693 | 0.02184 | 1.34888 | 30 | 1.68389 | 1.57389 | 0.11 | 6.53222 |

In this paper, a hybrid intelligent approach involving genetic algorithm for artificial neural network development has been proposed for voltage stability margin estimation in power system. The fast and accurate estimation of VSM has been considered as an effective way for assessing the stability of a power system from viewpoint of voltage. Implementation of GA makes it possible to achieve effective input-output mapping in artificial neural network with considerable speed-up in its training.

The proposed GABPNN approach sums up the goodness of evolutionary computing and artificial neural networks both. The value of this hybrid approach is that GA requires no gradient information so less susceptible than back-propagation to local variations in the error surface. Another advantageous aspect is that they operate in a population of possible solution candidates in parallel, instead of starting with a single candidate and iteratively operate on it using some sort of heuristics. The proposed approach provides acceptably good generalization ability during testing and found computationally efficient in VSM estimation. Successful application of GABPNN establishes the suitability of the proposed ANN model for online assessment of voltage stability.

The authors sincerely acknowledge the financial support provided by Department of Science and Technology (D.S.T), New Delhi, India under Research Grant no F.No.SR/S3/EECE/0064/2009 dated 22-01-10 and Director, Madhav Institute of Technology and Science, Gwalior, India for carrying out this research work.