Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 529724, 8 pages

http://dx.doi.org/10.1155/2015/529724

## Optimized Extreme Learning Machine for Power System Transient Stability Prediction Using Synchrophasors

^{1}State Grid Liaoning Electric Power Supply Co. Ltd., Shenyang 110006, China^{2}School of Electrical Engineering, Northeast Dianli University, Jilin 132012, China

Received 13 August 2015; Revised 10 September 2015; Accepted 13 September 2015

Academic Editor: Mohammed Nouari

Copyright © 2015 Yanjun Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A new optimized extreme learning machine- (ELM-) based method for power system transient stability prediction (TSP) using synchrophasors is presented in this paper. First, the input features symbolizing the transient stability of power systems are extracted from synchronized measurements. Then, an ELM classifier is employed to build the TSP model. And finally, the optimal parameters of the model are optimized by using the improved particle swarm optimization (IPSO) algorithm. The novelty of the proposal is in the fact that it improves the prediction performance of the ELM-based TSP model by using IPSO to optimize the parameters of the model with synchrophasors. And finally, based on the test results on both IEEE 39-bus system and a large-scale real power system, the correctness and validity of the presented approach are verified.

#### 1. Introduction

Monitoring the power system stability status in real-time has been regarded as an important work to guarantee the power system safe and stable operation [1, 2]. Up to now, the existing transient stability analysis (TSA) methods mainly can be divided into 3 classes: direct methods [3], time-domain simulations [4], and the extended equal area criterion method [5]. Unfortunately, these methods cannot work well for real-time stability analysis of modern complex power systems.

In recent years, pattern-recognition-based TSA (PRTSA) has been attracting the ever-growing attention of researchers all over the world [6, 7]. This kind of method has proved to be potential in the area of on-line dynamic security analysis by applying of the techniques of machine learning. By far, the PRTSA model mainly includes artificial neural networks (ANN), decision trees (DT), and support vector machines (SVM) [8–15]. However, the reported PRTSA approaches usually suffer from some inherent disadvantages and lack the ability of big data management and utilization, which restricts its further application in actual operating scenarios. For example, ANN has problems of overfitting, local optima, and slow convergence, and SVM has difficulty in parameter selection. On the other hand, wide area measurement systems (WAMS) provide the synchronous measurement information for the wide area power systems [16], which makes it possible to explore wide area protection and control schemes to avoid the system collapse [17–19].

In recent years, a novel machine learning algorithm called extreme learning machine (ELM) is proposed by Huang et al. [20]. Contrasted with those conventional PRTSA approaches, ELM has a lot of significant advantages, such as better generalization ability and a much faster learning speed [21–23]. Inspired by the social behavior of flocks, particle swarm optimization (PSO) algorithm is proposed in 1995 [24]. PSO has been widely used to solve a variety of optimization problems with many of advantages including good robustness, fast convergence speed, and high search efficiency [25].

In this paper, a novel ELM-based transient stability prediction (TSP) method using synchronized measurements is proposed. Moreover, to further improve the prediction performance, the ideal model is obtained by applying the improved particle swarm optimization (IPSO) algorithm to select the optimal parameters of the model.

The rest of this paper is arranged as follows. First of all, the used methodologies including ELM classification, PSO are presented briefly. Secondly, the proposed real-time TSP method based on IPSO-ELM is presented in detail. Finally, the proposal is tested using the IEEE 39-bus system and a real system.

#### 2. Related Methodologies

##### 2.1. ELM Classification

Assuming an ELM with hidden layer neurons to model data samples , it can be mathematically represented aswhere and are, respectively, the input and output weights vector, is activation functions, and denotes the bias of the th hidden node.

For the convenience of expression, (1) can be rewritten aswhere is the hidden layer output matrix.

ELM is to minimize the training error as well as the norm of the output weights [20]

Finally, the minimal norm least square method is used in the original implementation of ELMwhere is the Moore-Penrose generalized inverse of .

##### 2.2. PSO

The fundamental principle of PSO is to find the optimal solution in the complex search-space by moving candidate solutions (called particles) according to the competition and collaboration among particles through repetitive iterations. The movement of each particle is determined by a mathematical formulae over its position and velocity. In the iteration, the velocity and position renewal equation of th particle are, respectively, as follows:where and are, respectively, denoted as the local and global best known solution; and are, respectively, denoted as the evolutionary generation and the inertia weight; and are the learning factors, which represent the self-cognition and social-cognition in turn; and are uniform random numbers obeying the 0-1 distribution.

#### 3. Real-Time TSP Based on IPSO-ELM

##### 3.1. IPSO Algorithm

As pointed out by the famous “No free lunch” theorem, the overall performances of different optimization algorithms are equivalent [26], which implies that none of algorithms can always achieve the optimal for all aspects. In this paper, a mutation strategy is introduced to avoid the premature convergence to local optimum of PSO.

First, the optimization process is monitored by dynamically monitoring changes in population fitness variance :where denotes the population size, is the fitness value of the th individual particle, refers to the best fitness value in the whole population, and denotes the average fitness in the current iteration.

Second, when premature convergence occurs, a mutation strategy is used to maintain population diversity. Specifically, the positions of particles are updated by adding random perturbations timely, as shown as follows:where is the variation coefficient, rand is a real number randomly generated in the range from 0 to 1, and are, respectively, the positions of the th and th iteration of particles.

The criterion to determine the occurrence of premature convergence is given as follows:Here, and are the population fitness variances of the th and th iteration.

##### 3.2. Fitness Function

As is known, a proper fitness function plays an important role in optimization problems. In this paper, the fitness function is the classification accuracy of 5-fold cross-validation (CV): where is the model parameter vector to be optimized, which is represented by the position of each particle.

##### 3.3. Coding Scheme

A mixed-integer encoding scheme is used in the optimization process [27]. Here, it is considered that the th individual particle/state will be constituted by where is the total count of input features, () is a binary variable, and each binary code (“1” or “0”) refers to whether the corresponding feature is selected or not; () is an integer variable that defines the activation function of each neuron of the hidden layer as follows:

The use of parameters makes possible the adjustment of the number of neurons (if the neuron is not considered) and the activation function of each neuron (sigmoid or linear function). To facilitate the optimization, the decision variables are mapped into real variables within the interval ; and all variables need to be converted into their true value when computing the fitness value of each particle [27].

##### 3.4. Modeling Process

The modeling process of the proposed method can be divided into 8 steps.

*Step 1. *The used data preprocessing approach here is -score standardization method [15]:where is the mean of any feature in sample data, is the standard deviation of the feather ; is the normalized value corresponding to , .

*Step 2. *Initialization of the parameters: the maximum iteration number is assigned to 200, the population size is set to 20, and the number of ELM hidden layer neurons is 50.

*Step 3. *Initialization of the population: solutions are generated randomly, and each solution is corresponding to a particle, which is encoded according to (10).

*Step 4. *According to (4), calculate the output weights and the individual fitness values in turn.

*Step 5. *According to the optimization mechanism of PSO algorithm, update the location of individual particle and generate the next populations.

*Step 6. *Dynamic monitoring of changes in the population fitness variance: once premature convergence occurs, save the current optimal solution , and go on to the mutation operation. If a better solution () is found in the solution space, then update the optimal solution , and quit the mutation operation.

*Step 7. *Judgment of termination condition: the optimization process will be terminated, if the current number of iterations exceeds the prespecified maximum number of iterations or the value of fitness function is greater than 99.00%; otherwise, and jump to Step 4.

*Step 8. *Acquisition of the ideal model: output the optimal solution , and obtain the ideal TSP model.

The flowchart of the modeling process is shown in Figure 1.